What tool(s) should I use to make a hole in a wall and then put a screw in?
Like StrongBad mentioned, it depends on what your walls are made of. If you're in the US, they're most likely drywall on top of wood studs, in which case you either drive screws directly into the studs, or use anchors to secure the screws in the drywall itself. Studs are the best option, but they rarely line up with where you'd want them to be. For a kitchen utensil rack, anchors should be fine.

Tools: depending on the anchor type you choose, you'll probably need a drill (as an apartment-owner, a 12 or 18V cordless drill/driver combo will be invaluable) with a drill bit to drill the hole that the anchor goes into, and a Phillips-head screwdriver (or screwdriver bits for your drill) to drive the screw into the anchor.