if you buy plasplugs in a block, you will find that the joining strip has samples holes in it, and embissed on the strip are drill and screw sizes.
To prevent purniture falling over when you climb on it, I'd use brown plugs and 40mm screws. They might be a bit excessive, but you want strength and safety.
if the wall is brick or block, you want masonry drillw, which are silver, with a nearly-square "point" which is tipped with a piece of tungsten carbide brazed on. They are usually silver.
If the wall is plasterboard on wooden studs, you need to find the studs (ask how) and you need to drill through the plasterboard into the stud with an HSS drill, usually black. No plug required for wood.
You need a hammer or combi drill (but not SDS unless you will be drilling into concrete quite often) .
For indoor DIY work a 12mm chuck, mains or cordless, will do. If cordless, get an 18volt one. Own-brand from Lidl or Wickes is fine. a big heavy drill is not what you want. Nor is a cordless drill/screwdriver, which is too weak. Some drills come in cases with a starter set of bits and drivers. Two batteries are very useful with cordless. Mains will be cheaper and more powerful.
this sort of thing
or this
will do for your first.