I'm currently playing a Dragonborn Cleric, which has the Life Disciple domain.
I was wondering if there's any other classes that could be good to combo with this cleric? I just reached level 5 now though, and I don't know if there are any rules about multiclass level, if it can only be done at level 3 or if you can do it at any level up?
I wouldn't recommend multiclassing as a spellcaster unless you have a specific goal in mind. You're delaying access to your higher level spells to have a wider variety of lower level spells. Getting higher level spells sooner is usually better.
Spellcasting classes that use the same spellcasting ability go well together since you don't have to spread your ability scores thin. Druid would be the obvious choice, since they're the only other full spellcasting class that uses wisdom, and Disciple of Life works well with Goodberry.
The rules for multiclassing are in the Basic Rules. If your DM allows it and you meet the pre-requisites, you can multiclass at any time past level 1.
If you game as Feats you can grab some of the great parts of what InquisitiveCoder mentioned about multiclassing Druid without losing class progression. You can take Magic Initiate: Druid. As a life Cleric taking Shillelagh as a cantrip is great as it allows you to hit/damage with Wisdom while using a club or a staff for 1d8. Also you can take Goodberry and get 1 casing per day. Each berry heals 4hp, instead of 1hp due to Life Cleric. http://dnd.wizards.com/articles/sage-advice/rules-answers-august-2015
I will point out that multi-classing isn't terrible for a caster if you only take a level or so, especially if you take a multiclass as another caster. As long as the other spell caster isn't Warlock you lose as quick access to high level spells, but you don't lose access to those spell slots.
Example: A 1 Cleric/2 Wizard. Can't cast any lvl 2 spells, but it does have 2 lvl 2 spell slots, and so can cast 2 level 1 spells with a level 2 spell slot.
Thanks! At level 4 I actually did pick Magic Initiate, but I decided to pick my own class again because I’m more of a ranged Cleric, I’m usually never close enough to touch or hit the enemy with a mele attack. And because of that I figured it be nice to be able to cast one more extra buff, healing or ranged attack!
I did not think of the Goodbery though, that would indeed be a nice thing to have to heal! Then again, I’m new to D&D still, and I’m just learning! I will keep that in mind for a different time though!
There are some levels that are generally viewed as better times to multiclass than others, but that doesn't matter too much if you're not min-maxing (trying to be as powerful as possible). On that note, there's generally not many options to become more powerful when multiclassing as a Cleric. However, there are some fun options such as taking a level or two in the Monk class.
The most popular multiclass including Cleric is being a Ranger or Druid, and taking 1 level in Cleric with the Life domain. This is normally for the sole reason of making your Goodberry extraordinarily powerful.
It could be fun to try it in reverse and take a couple of Druid levels while mainly being a Cleric, however. While you delay access to your high level spells, you do not delay access to your high level spellslots since Druid is also a full-caster class - this allows you to cast your lower level spells with the higher level spellslot if they scale (such as Guiding Bolt or Cure Wounds). You could put two levels into Druid and gain Wildshape for some utility/situational movement, new lower level spells, and a 2nd level subclass feature - the Circle of the Shepard subclass lets you put down a totem as a bonus action that buffs nearby allies, which might be fitting in a support role such as the Life Cleric!
However, if you're new to D&D it may be advisable to stick with one class for a little while longer - it depends how confident you are with the current game mechanics you know.
On a final note, another cool option with Magic Initiate is to choose Wizard and take Find Familiar. When you have a familiar, you can cast touch spells such as Cure Wounds through the familiar rather than yourself, greatly increasing the spell's potential range with some clever positioning and its use of the ready action for movement.
Just to get a bit more info what are your stats as that will drive what else you could possibly multi class into. Cleric can easily multi class into a number of other classes depending on what your stats look like, fighter being one of the easier ones. Bards are another fun option if you have the requisite charisma to be able to.
Also did you choose cleric with your magic initiate to get more cleric cantrips or just a free daily cast of a spell? I only ask because you could get cantrips and spells you wouldn't normally be able to access if you chose another class.
I will not do multiclassing I’ve decided, for that reason you mentioned; I’m new to the game and I think it’d be a little too advanced for me at the moment.
However, I had no idea about the Find Familiar thing you could do with touch spells! I will definitely have to keep that in mind. Do you know if I’m able to pick Magic Initiate again, on the next level I get to choose a feat? Or can you only pick one of each feat?