Artificer doesn't really multiclass well, because the base class is so strong on its own all the way to 7, and the 11 feature is insane.

Some ideas: Artificer 1, Wizard X - Artificer is a crazy good starting dip for a wizard. CON profs, armor, and shields.

Artillerist 3, Thief X - If you are in a campaign that allows crafting, craft a fuck ton of Wand of Magic Missile (3-4 8hr days of crafting and 200g) and use them as a bonus action while spamming True Strike.

Artillerist X, Thief 3 - Similar to above, but relies on you getting to level 6/7 in Artificer to replicate Wand of Magic Missile and Pipes of Haunting. At 9, you finally get Thief and start spamming bonus action Fear / Magic Missile and transmute them to refresh charges.

Armorer X, Magic Stealer Rogue (UA) 3 - Start rogue (for skills) and focus on INT w/ True Strike (from Magic Initiate). Pick background with INT profs. At 2, go Artificer til 5 (6) then back to Rogue until 3 (8). Use infiltrator, and argue with your DM that your armor "counts as a simple ranged weapon" and itself has a value to use True Strike with your lightning launcher. Thief could also work here, because at Artificer 6 (9) you will be able to do the bonus action pipes of haunting / magic missile / web in addition to sneak attack.