What do I have to write down in my notebook? I'm a new DM and I want tips!

You can make note of future encounter ideas, treasure coin, gems, art object, magic items, important non-player characters, plot ideas, important details of  past interactions or actions, the party reputation, watch tour and marching order, characters initiative bonus, passive perception and other features, random encounters, rumors, legends etc

It varies from person to person, but some of the basics:

Any proper names you give the party, be it a place name or a random bartender, and with the name write a short description of the context for who or what the name refers to... Players will ask about it eventually.

Write down your players' hit points, AC, Saving Throw modifiers, Passive Perception, and Passive Insight.  This lets you keep exploration and combat more cinematic, not having to ask the players for this info every time something happens.  For example, a trap, you can use their Passive Perceptions if they aren't actively looking for it.  If they're talking to someone who might have less-than-benevolent intentions, you can use their passive insight to determine what or how many clues you give them for this individual's true nature.  And knowing their combat stats, along with about how much damage they put out in an average round makes it much easier to balance encounters (CR is not a good gage.  You have to look at special abilities and action economy.  For a Tier 2 party, a dozen CR1/2 creatures of one type is a joke, but of another type could be a deadly encounter.  It all depends on if the players' damage output is enough to kill or cause-to-flee all of the enemies before the enemies kill all or most of them.

Any lingering effects on party members, or magic items you gave them.  This will help you gage how combat-ready they are when planning your next session.

You will find that some players forget to mark off hit points when they take damage, or mark off a potion they used... and it might be innocent or deliberate.  When such a player is at your table, you may want to have a simplified version of a character sheet with their spell slots, hit points, and expendable items, just to keep everything honest.

I try to write out the plan for the session before hand, and if you do the same, double-spacing, you have room to cross out anything that ended up not happening, and writing in what you had to improvise due to the party's inexplicable decisions.  This gives you a running track of what's happened, and lets you see what sorts of hooks the party goes for, and which ones don't work with your current group, helping you better plan for what will happen without resorting to railroading them.  Remember, anything they don't encounter, you can pull aside and save to introduce later so that prep-work isn't wasted.  You may need to tweak it to fit where you later find a place for it though.

FayetteGamer gave some great tips. The only thing I'd add is to write down anything your players do in RP that you think might be cool to follow up on or revisit in future sessions. 

For example, in my last session, the rogue mentioned wanting some pastries. I made up sesame honey buns on the spot, and the rogue's RP made them sound life-changing. So I wrote a note to myself to have the sesame honey bun cart guy become a recurring NPC. Just a small worldbuilding detail that brings a smile to my players. Other notes on player choices have included things like, "Cleric willing to sacrifice for power - maybe lycanthropy arc?" and "artificer drank magic water, ask for CON saves versus wild magic effects." Sometimes the story can take cool turns or involve fun callbacks when the DM is attentive to player actions and potential consequences. 

Some of the main things I note down:

1: Consequences. If a player decides to break into a shop at night because they can't wait until morning, you'd better be sure the "party bounty" tab of my notes is getting opened! Write down anything that they do that you think can have ramifications in the future - helping someone they may need in future, killing guards, clearing goblins, flirting with the barkeep, letting a bandit escape. Anything you can bring up again to make them remember what they did and feel like the world remembers their passing, rather than being a pregenerated and stale world where they could be anyone and do anything without any difference.

2: Theories. When a player says "Ooh, what if >blah< is actually >bleh<?". In my game, this was "huh, we have a letter from the bank saying we have paid for the school we were building. I bet that was Enzo, he'd do that for us." - (spoiler - it wasn't!). Noting down player conclusions and suspicions is very useful for making what they are paying attention to relevant.

3: Power Moves. making a note of what the players use often is a good way to twist the combat up next session (or this one, if you're good at thinking on your feet). 

4: What players are looking for. I have a character who is desperately trying to get stronger, despite being str 20, so I am dropping little things in there for them - a Whey Spider that gives you +2 Str if it bites you (but is addictive), a belt of giant strength (str23), potions of strength - and he keeps turning his nose up at them, which I can't quite fathom. Having a list of things the characters want is very useful when you need to link a random encounter to their adventure - a random dungeon with loot is awesome, but a random dungeon with loot and a magical instrument that the Bard has been trying to find is awesomer!

5: Their effects on the background plot. In my campaign, things are happening that the party do not yet understand. They have scuppered a ritual, they have fought a giant snake, killed a dragon, and made enemies with a giant king. Things all seem unrelated, but as the DM I know better! So each time they do something, I make notes on how it affects the plot in the background. Most recently, they destroyed a giant skyship from the giants armada. I need to note down what effect this has on the world.

6: NPC Names, You make them up on the fly, and it's important to keep notes so the world is congruent. Don't be afraid to say "I've forgotten what I named her so she is, and has always been, called Glynda". better that than ahve them pipe up halfway through roleplay that they ask why her name changed.

Something I've started to do with note-keeping is writing down any voices or accents I use with characters, to help me keep things consistent. I've finally had to accept that I just won't remember on my own-- especially if it's an NPC I made up on the spot

I started using a website called Kanka.Io

It lets me create, basically, a wiki based around my campaign.  Every NPC, every location, gets an entry and then I do a journal entry after every session to document who went where and what NPC's they interacted with.  I also have factions in there that I'm keeping track of which NPC's are members of and it's all linked up so I can quickly get from entry to the next.  I like it but without paying for it (like $5/ month) the ads are OBNOXIOUS....

I also use it as a patreon reward in that supporters get private access to all these notes (I keep DM secrets hidden) but it helps provide a tool for players and myself to quickly skim through to recall details.