I’ve got another character concept to present, but this one I’ll actually be playing! We’re starting at level 3, and I have no idea to what level we’ll be going.

A Rock Gnome Artificer-Artillerist/Evocation-Wizard. I’m reflavoring the Eldritch Cannons to be improvised “wands,” and my starting level is artificer for those CON saves.

My conundrum is this: Do I go for a level balance of 3/X or 6/X of Artificer/Wizard? Only taking 3 Artificer levels would get me the cannons at starting level, and then I could pour the rest into Wizard. But taking 6 levels would get me an Arcane Firearm, Tool Expertise, and an extra Infusion Slot at the cost of delaying my spell progression. I’m honestly quite divided on this, what do you think?

EDIT: A decent part of me is considering going for War Wizard. I was originally planning on taking Evocation Wizard to reduce spell copying cost and sculpt my evocation spells. But the more I think about it, I kind of want to go with War Wizard to increase my defense and damage. Abjuration is also be a decent option.
EDIT: I’d like to stray away from Bladesinging, simply because I am not a melee build. I recognize it has some nice, non-melee boosts, but I’d rather use a chain shirt and some other defensive features. My main plan is to use an Enhanced Arcane Focus to cast spells in one hand, and an Eldritch Cannon in the other. (Yes I know the cannon can be on the ground with legs. But the flavor!) I’d use the Flamethrower and Force Ballista at early levels, then shift towards Protector as my 2d8 damage cannons begin to be outclassed.

Personally I'd either stick to 3/17 or go all the way up to 9/11 or potentially even ditch Wizard completely.  6 is giving you a few bonus features but nothing fantastic, delaying your spell progression and stopping you short of amazing class features such as flash of genius, your ASI, 3rd level spells and your Explosive Cannon ability.  If you decided to stop at 6 because of the Evocation Wizard's Overchannel ability at level 14:

1. You wouldn't actually get that until character level 20 anyway if you pick up your 6 Artificer levels right now.
2. You still won't be able to get anything higher then 7th level Wizard Spells and not until character level 19.

This is the reason why most people seem in agreement with full caster classes not dipping more then 3 levels into other classes, getting up to 9th level spells anddelayed spell progression.  There are exceptions of course (Sorcadin being a prime example) but that plays differently and uses the spell slots as fuel for other class features instead of actually casting high level spells.  Thus, I'd ask myself, do I want to play a caster or do I want to play something else and augment my existing abilities with spells/spell slots.  The answer to that question should help you take your pick.

If you want to be more of a caster, I'd stick to 3/17, otherwise or for a more utility-oriented character, 9/11 is a great stopping point though you'll never get access to spells higher then 6th level on the Wizard's spell list bearing in mind that your progress will sort of stagnate from character level 9 through 16 when your Wizard levels will finally grant you 4th level spells to play with compared to a full-blown Artillerist Artificer who gets many neat features throughout their progression (Magic Item Adept, Spell-Storing Item, Magic Item Savant, Fortified Position, Magic Item Master and ultimately, one of the best if not the single best capstone in the game, Soul of Artifice).

I'd start at Artificer 3 and begin taking wizard levels from there. See how things go, how your party works and stuff like that. You will be able to make a much more informed decision after several levels of play in the campaign.

Certainly you can pick whichever official plan you like (go ahead and make versions of each in the builder if it's fun for you), but keep your eyes open and be willing to deviate from it based on how the game goes.

Don't try to plan to 20 if you have no idea how long you'll be playing (and even if you do, it's not a good idea to plan that far ahead). Take levels in line with where the story is leading you, what you feel the party needs at the time, and how you get along with the classes you take. You'll end up with a better rounded, unique and more interesting character that way.

So, here's the thing about both Artillerist and Evocation Wizard... both classes get access to an always-on damage boost to their spells eventually, but those damage boosts are also mutually exclusive... Arcane Firearm only grants additional damage specifically to Artificer Spells, and Empowered Evocation only applies to Wizard spells. So with that in mind, your best bet is to prioritize one class over the other... if you intend to be primarily a Wizard, you're not going to get much use out of Artificer past level 3 until you hit around level 7. You won't really get the most out of an Arcane Firearm if you're mostly casting Wizard spells, and tool expertise is nice, but if your focus is on crafting wands... crafting is mostly a question of time and resources, with the actual roll on crafting being less important. The main thing you're missing out on is infusions, and although those are nice, you can get similar results through smart spell selection as a wizard.

That said, I, personally, prefer to minimize multiclassing as much as possible, and I actually think there's more value in going mostly Artificer with Wizard as your bonus. Wizard pays off faster with useful traits at earlier levels... you hit the Evocation Subclass at only level 2, and all of the features you gain don't step on the toes of anything you get from artillerist. You'll only get access to level 1 Wizard spells, but you still get access to a spellbook which you can fill to the brim, and Wizards get a ton of great utility spells at level 1. You also gain the ability to ritually cast any spell you have in your spellbook without needing to prepare it, so if you've got 10 minutes of in-game time you can always whip out Identify or Detect magic, and you can even nab Find Familiar and always have access to it, regardless of whatever else you do for yourself in the adventuring day. Meanwhile, this frees you up to go whole-hog into Artillerist, where, if your game takes you all the way to level 20, the only unique feature you'll miss out on is your capstone Soul of Artifice ability.

Quote from 54MU31 >>

I’ve got another character concept to present, but this one I’ll actually be playing! We’re starting at level 3, and I have no idea to what level we’ll be going.

A Rock Gnome Artificer-Artillerist/Evocation-Wizard. I’m reflavoring the Eldritch Cannons to be improvised “wands,” and my starting level is artificer for those CON saves.

My conundrum is this: Do I go for a level balance of 3/X or 6/X of Artificer/Wizard? Only taking 3 Artificer levels would get me the cannons at starting level, and then I could pour the rest into Wizard. But taking 6 levels would get me an Arcane Firearm, Tool Expertise, and an extra Infusion Slot at the cost of delaying my spell progression. I’m honestly quite divided on this, what do you think?

Tool Expertise will come up almost never, but being 6th level does unlock new infusions for you, allowing your battle loadout to be +1 armor and a +1 shield (due to repulsion shield).  Your third infusion is highly likely to be a cloak of elvenkind or bag of holding.  The Arcane Firearm's utility depends on how your DM interprets its word salad restriction of you casting a spell through it, which doesn't actually mean anything.  For example, do you need a free hand to cast a spell through it?  We literally have no rules for it and the answer has a huge impact on how useful the ability is.

That said, you're mixing Artificer with Evoker, not Abjurer.   There's just not a whole lot of synergy here (e.g. even if you tunnel to L5 for the Firearm, you won't get the bonus to Magic Missile).  Might I suggest a radically different approach?  If you start off as a Hexblade 1/Evoker 2 (statline with Point Buy Int 16/Dex 14/Con 14/Cha 14) you'll have medium armor proficiency and, critically, Hexblade's Curse, which has the synergy with Evoker Artillerists don't have. 

If you want to lean into Artificer/Wizard, Artillerist/Abjurer, Armorer/Abjurer, and Artillerist/Bladesinger all have interesting synergies.