I'm designing a campaign, set in a world where every human is a wizard. Since my players are a big fan of using magic in DnD, they're very excited to start this adventure. However, I'm having some troubles preparing this campaign and being a new DM, I sure could use the help of some more experienced members of this community.
Apart from the class being limited to only wizards, most of the standard DnD 5E rules apply during the campaign.
I have 4 players starting at level 1 and my goal is to end the campaign when they're level 7.

Question 1: All the players will be wizards, so in combat they will have to rely almost exclusively on magic. I was wondering if the number of spell slots, suggested by the Player's Handbook (PHB p.113), will be sufficient. I'm a little bit worried they will run out of slots quickly and their cantrips won't do enough damage to defeat the enemies.
I'm interested to hear your opinions on this and if you maybe have a solution to this problem.

Question 2: Since every human in this world is a wizard, the players will often run into enemy wizards. I'm having a very hard time balancing these enemies. I usually rely on CR to determine the strength of the enemies, but I can't find a connection between CR and an enemy wizard's level. I'm completely lost on this... I looked in the Dungeon Master's Guide, but couldn't find any rules on creating enemies with magical powers (maybe I missed them?). I did manage to find some spellcasters in the Monster Manual, but the CR given to them, doesn't seem to be consistent.
For example: A Druid (MM p.346) is a 4th-level spellcaster and gets CR 2. A Priest (MM p.348) also gets CR 2, but is a 5th-level spellcaster.

I really could use some help on this so I can continue creating this campaign.
Thanks in advance for any info or tips!

Have you considered playing the Strixhaven soucebook that plays at a magic school / campus? That one is kind of catered for magic users and with players playing magic users predominantly. Would be an easier way into DM'ing for a first time DM than crafting your own campaign.  

Q1: I might not worry about spell slots and character levels in combat. I find that focusing on balancing the encounter to the party works just fine. You might find SlyFlourish - Dials of Monster Difficulty helpful, in the event that your encounter build is heavy-handed, or turning to a handwave. Some out there believe that the "problems" in an encounter are the DM's responsibilty to solve, I disagree. The DM provides obstacles and challenges for the party to overcome. Solutions are a function of the party, let them come up with what they can.

Q2: You might find some inspiration in making humans a vast minority to other races. The NPC statblocks don't *have* to be of a specific race. You can have goblin bandits, or kobold acolytes all the way to dwarven archdruids. Maybe use the statblocks as template to create the creatures that you need to have. From your description, I'm assuming that you have created PCs as enemy wizards and are trying to convert them to NPC's CR. It will be a bit difficult and requires some math and chart lookup. (If you ever played 1st, 2nd, or BECMI you'll be familiar.) Head over to Ch. 9 - Creating Quick Monster Stats. Follow along with the step-by-step until you come up with the average CR of your monster. If I have a creature that I've heavily modified or created by PC-sheet, I usually skip straight to Step 4. - Final CR.  You might find the most "difficulty" with spell casters as some of their abilities can overlap. I might suggest to set up a plan on how the monster would act in combat. Spell casters will have limited time to get anything off, so work from most damaging/controlling down. Work out the DPR over 3 rounds and at the end, you may just wind up shooting from the hip. In which case, I might suggest leaning on the high side of CR, as this will prevent your overall encounter balance from being heavy-handed or overbuilt for the party.

Thank you both very much for the quick reactions and the helpful information.

@ Kaavel: Your detailed explanation will help me a long way. I believe I might be able to continue creating my adventure thanks to you.

@DMGorani: I will definitely check out Strixhaven and see if I can use some of its mechanics in my game. 

Maybe encourage one of them to be a bladesinger, so there’s someone up in melee to give the others some breathing room.

The other thing I’d worry about is healing. Make sure you give lots of potions, or encourage someone to take the healer feat, or maybe even chef, or homebrew cure wounds into a wizard spell.

Regarding the spell slots, I don't think there will be a problem. Most monsters that they will reasonably face at level 1 and 2 in numbers can ne easily taken down with cantrips and with the way damage scales with spell levels, the stronger ones should be taken down easily with combined effort. The issue might be when you get to the levels in which you have many enemies with more health but that can easily be handled with area attacks.

If you start to notice that they run out of spell slots though, you can also give scrolls and staffs as reward or give them downtime for them to craft their own scrolls.

One thing you have to consider though is that being a wizard can get very expensive between getting spell components and transcribing spells to books. You will have to balance the rewards so that they can actually buy the materials they need but not too much or they will cheese the game by having each choose a different spell list and then use money to copy the spells from each others books to get around the known spells restrictiond.

Quote from Farenas >>

One thing you have to consider though is that being a wizard can get very expensive between getting spell components and transcribing spells to books. You will have to balance the rewards so that they can actually buy the materials they need but not too much or they will cheese the game by having each choose a different spell list and then use money to copy the spells from each others books to get around the known spells restrictiond.

Or, in a world where everyone's a wizard, just reduce transcription costs because everyone needs that stuff rather than the inks and such being a specialty item

A thought, which  you may take or leave, from a DM who has for  several years run a campaign in a world set in an empire whose aristocracy is composed of wizards.  Remember that wizardry is "book learning," at least as written, so you are working with a society with a fairly good educational system of some kind, formal or informal, where everyone picks up at least the basics of wizardry: a few useful cantrips or low level spells, anyway.  How far does everyone else go in education?  I assume a farmer or craftsman has a primary education, but they have other things to learn as well.  Likewise not everyone has the aptitude or interest or financial wherewithal to pursue high-level intellectual work.  Even with a basic education it seems likely that some will be pulled into other paths, especially among such exceptional people as are called to go on adventures: called to follow a god or gods, called by the arts, inspired by ideals of chivalry and purity to take up arms and right a fallen world, spurred by necessity to make a shady living by stealth and wits...  I guess what I'm saying is that I'm not sure you have to be as restrictive as you seem to want to be - talk to your players and see what they are interested in, but even a world where everyone  has some magical education, (represented, perhaps, by a free magic initiate feat?), not everyone need be a full wizard by class.  Even if they do all start out as wizards, you may want to be generous with opportunities for multiclassing. 

The reason I suggest this is that the all-wizard party is likely to be hard to run for for a new DM, unless the campaign is very focused on politics, interpersonal conflict, and roleplaying, to the near exclusion of direct combat.  The party is missing a lot of the usual roles, and the game assumes by default a fairly wide mix of roles and abilities.  An all-wizard party in particular is fragile, and you will have trouble gauging CRs.  (An all-cleric party probably works best in terms of a one-class party, though a case could be made for others). Characters in such a party will have to be very smart and strategic about handling combat challenges, which may be hard for new players, and may bog the game down into endless planning.  Unexpected encounters may go badly against such a party, and you may find combat very "swingy" in the sense that a bad initiative roll or a monster making a save or two can mean the difference between an easy victory and a TPK, with little middle ground.  You may also have trouble giving each player "spotlight time" - time where their abilities in particular save the day in some way - because different wizard subclasses are not that different from each other, compared to the differences between classes.  Keeping everyone entertained will therefore be more of a challenge than usual.  I have run "magical academy" games where everyone is a wizard - but not in D&D, which isn't really set up for it well.  Not that it couldn't be done, but just a warning that you are setting yourself and your players a stiff challenge, and perhaps not one  you have to take on, even if you want the feel of a world in which pretty much everyone knows some magic. 

It can certainly lead to interesting world details: the library has 15' high bookshelves but no ladders, because the assumption is that everyone who can read, at least the books on the upper shelves, knows mage hand.  There are no laundresses - everyone cleans their clothes by prestidigitation.  All housekeepers simply use unseen servants for gruntwork. No point to armor if all soldiers have wands of magic missile. Universal knowledge of even a few cantrips and first level spells could lead to a society radically different from your typical  D&D "kinda-sorta-late medieval" setting.  You've set yourself a stiff world-building challenge as well.

With an all wizard party you are probably going to have plenty of spell slots. I would actually be worried about them having too many as wizard game play is largely about managing resources and collectively they will have a very big pool. I think you will probably have players trying to take long rests allot and you not make that too easy, the game is not balanced for wizards being able to burn their entire load of spell slots in every encounter. You do need to wear down their supply and make them use spells wisely otherwise it will become impossible to deal with the shenanigans of constant teleportation, explosions and mind control. 

In terms of encounter balance there is no real connection between cr and caster level. Cr depends on the damage they out out, they're likeliness to hit and alike.  In that sense a class doesn't matter its about the abilities and spells are really just abilities that use spell slots as a resource. You need to look at the spells and balance them like normal monster abilities, a fire ball is just a dragons fire breath in a sphere instead of a cone. If you are home brewing my advice is don't actually run high level full casters as enemies that much, its too many spells and slots to keep track of and can really slow you down. Instead pick spells and use them like monster abilities then calculate the cr based on the damage and likeliness to hit of those spells. For example you can have a monster whose attack is fire ball and when he runs out he does something weaker like fire bolt (combat rarely lasts more than 3 rounds). Faster combat is better and you are better off spending time describing how they throw a fire ball than mulling over which spell to use. There are also allot of magical creatures that don't cast spells make sure there is combat variety.

Now, what might players actually be lacking in an all wizard game? money and skills. They need ways to get them but again, never make it too easy. My advice is actually to do things like offer alternative paths to get what they need. Things like a side quest to rescuing a shop keeper that unlocks a new shop with what they need or dungeons with a rare spell on a stone tablet or an expensive spell casting ingredient.  You can give knowledge of spells and casting items as rewards from dungeons rather than always making them buy them. Do not just give them extra gold because they're wizards, players being too rich is allot worse than too poor they'll still get some spells through leveling instead see them needing these things as an opportunity to create quests and challenges.