Charbs nailed it. The craftsman kits are a great relatively inexpensive way to get a fairly decent starter mechanics tool set.
Things not to buy Chinese:
-Wiss snips
-Channelocks
-Vise grips
-rachets
-any air tool
This too.
And in no particular order:
I'll add that having a bench with a vice on it - and preferably bolt the bench to the floor - is a really nice option if you have room. I just have some basic stuff made from 2x4's and plywood, works great.
Also, those yellow 500 watt halogen lights everyone sells are fantastic. Got to see what your doing.
A jack and jack stands, of course. I know a lot of people don't trust Chinese castings, but I've had excellent luck with my harbor freight 6 ton jack stands.
I personally never had any luck heating rusted bolts etc with a mapp or even a mapp-oxy torch. I got to the point where I thought the whole "heat the bolt up" thing was kind of overrated... until I bought an oxyacetylene rig. That was just a $100 harbor freight torch [plus the cost of cylinders], worth every penny.
I use taps more then I ever expected I would. A good set of those is handy to have around, but not a out-of-the-gate priority. Think Irwin or Vermont American.
Consumables like, rags, chip brushes, wd-40, nitrite gloves and fast orange hand cleaner are nice to have. If you don't already, dedicate some old clothing as work clothes.
I know you're just talking about hand tools, but be mindful of PPE for whatever you're doing, in particular Z87.1 safety glasses. Yes, they look dorky, wear them anyway. You get two eyes for depth perception, not redundancy.