I'd highly recommend a toolbag with a shoulder strap. Makes working on projects around the house a breeze and you're not running back and forth because you forgot something. Below are my recommendations for getting started with links. There are cheaper alternatives to everything, but these are all tools you'll quite often as a homeowner and I follow the "buy once, cry once" philosophy.

• Ratcheting screwdriver
• Standard screwdriver set
• Socket set imperial & metric
• Multi-function Needle nose Pliers
• Cobra Pliers
• Adjustable Wrench
• Allen wrenches
• 24" level - long enough for accuracy but not cumbersome to haul
• Claw hammer
• Utility Knife
• 5 in 1 tool & putty knife
• Studfinder
• Non-contact voltage tester - make sure the power is off!
• 18/20 volt impact driver/drill combo set - either Dewalt, Milwaukee, or Ryobi are decent brands in the US. There are better and (a lot) worse brands but those are all quality brands for homeowners. Pick a brand and stick with it so you only need one set of chargers/batteries.
• You already have a tape measure, make sure it's at least 25'
• a bag to carry everything - This is what I have, but there are many options out there.

And since your original question included picture hanging... buy a picture hanging kit with a bunch of hardware. You'll need it for years of hanging and you'll be happy you don't have to run out to buy each time.