I want to build a wizard with the classic thief skills: pick pockets, open locks, find/remove traps, etc.
what is the best race, classes, background, feats combination to do this? Am I better off taking levels in thief, then wizard or wizard with backgrounds/feats to accomplish this? The character will be level 7.
Honestly, starting with 3 levels artificer would be great, but it is still UA. Gets you thief's tools proficiency and expertise at level 3, plus medium armor and CON save proficiency for concentration.
Rogue is a good secondary option.
I made a locksmith character once that was a level 3 artificer mark of warding dwarf. With 16 DEX and guidance (artificer spell) I had a +2d4+9 to all rolls that involved locks. At level 3.
so we're looking to gain the skills: Investigation, Stealth, Sleight of Hand and Thieves Tools?
Background wise I think Urchin would be the best fit but Charlatan or Criminal would also be good choices
Race Wise pretty much any race could be beneficial, but I would go with either Elf, Variant Human, Light-foot Halfling or Svirfneblin Gnome:
Elf will give you some weapon skills to help give that thief image and extra magic either through cantrips with High Elf or Drow or Wood Elf Magic but you may need to invest in some extra feats, Drow High Magic from Xanathars guide makes Drow particularly useful (you can get Dispel Magic and Levitate both x1 per LR and Detect Magic @ Will at 4th level) but Wood Elf can get a Pass Without trace at 4th level as well. Variant Humans are always good for a free feat, taking skulker or Dungeon Delver could be a good thematic fit, and Light-foot Halflings are you go to race for easy hiding, as a little bit more of an exotic twist you could go Svirfneblin Gnome, if you are prepared to invest in a feat at 4th level you can get Svirfneblin Magic which gets Nondetection at will with no material components required which means you are immune to all divination magic whenever you want.
Class wise: I probably stick with single class wizard as the only other skills you need for the wizard/thief build would be Investigation and spells may be more beneficial than taking up to 4 levels of rogue. All wizard schools have their merits but for a wizard thief I think its between Divination, Enchantment and Illusion.
Feats wise: as previously mentioned, Skulker and Dungeon Delver could help with hiding and finding traps, but also Observant for that highly dubious +5 bonus to your passive investigation/Perception and lip reading.
Spell wise, I like Shadow Blade for that stealthy kill but you've also got access to various teleportation and movement spells (jump, longstrider, misty step etc) and that amazing spell Find Familiar, it's your partner in crime, your extra set of eyes, your inside man and early warning system. If its a 7th level wizard you'll also have Greater invisibility to help get around unseen or Dimension door to get away or confusion to deal with that unexpected guard. Gust, Mage Hand, Prestidigitation and Minor Illusion are all useful cantrips for creating diversions but means you may have to make a choice between thematically appropriate cantrips and having some easily repeatable damage dealing potential. A word of warning about the spell Knock...it opens any lock, which is great, but a lot of player miss the very last line of the spell: "When you cast the spell, a loud knock, audible from as far away as 300 feet, emanates from the target object" so if you are planning on going down the wizard thief route maybe best to stick with your thieves tools.
My personal build for this would be geared more towards stealing/breaking and entering:
Background: Urchin (gives you Stealth, Sleight of Hand and Thieves Tool proficiency as well as disguise kit which could come in useful),
Race: Svirfneblin (gives you bonus to both Dex and Int as well as 120ft dark-vision),
Class: level 7 Wizard (Divination, for those sweet portent dice and getting additional spell slots back),
Feat at level 4: Svirfneblin Magic (gets you Nondetection @ will and Blindness/Deafness x1/LR, Disguise Self x1/LR and Blur x1/LR)
Cantrips: Mage Hand (although bear in mind the hand is a visible thing), Minor Illusion (for creating places to hide), Mending (for when your lock pick breaks or you need to repair that ornament you accidentally knocked of the shelf whilst climbing in through the window) and Prestidigitation (for cleaning the mud/blood of yourself before going back into a public area)
RAW sneak attack requires a finesse weapon or ranged weapon so won't apply to spell attacks although that said It could apply to a booming blade or green flame blade cantrip though as they use a normal attack with a weapon not a spell attack.
High Elf for dexterity + intelligence boost, and free cantrip.
Goblin for dexterity + constitution (both useful), Bonus Action Disengage and Hide (like a rogue, without having to be a rogue!)
Gnomes can be good, too...they have a racial feat that's good for wizards & thieves that let's them turn invisible if they take damage. They are also naturally intelligent...and sneaky.
I'd say Urchin background personally, for a situationally useful skill in cities, skills, and the flexibility to tell your story.
Thief Rogues can use items in interesting ways, and still cast spells in the same turn...lots of interesting combinations. Mix with Bladesinger, Illusion, Enchantment or Divination Wizards for sneaky thief action!