It's funny how little you can actually get by with starting out. Most forget that after accumulating all the specialty tools (guilty myself in some recent times). In between jobs currently, so all my 'shade tree' work is done out of a 3 drawer Husky box with the basics and a Craftsman bag with some electrical and specialty tools (including some M12 cordless impacts, once you spoil yourself with those, it's hard to go back). I'm sure my list will be about the same as the rest, trying to remember what I had just starting out.
1. Socket set - 1/4" & 3/8" chrome set with extensions and ratchets, 1/2" impact (planning ahead, I stopped using my 1/2" chrome sockets when I switched to impact, still use them on a breaker bar though)
2. Wrench sets - Don't need to the biggest set starting out (I think I only had up to 3/4" and anything larger I just used an adjustable)
3. Hammers - Mini sledge, ball peen and dead blow (Still the only 3 I use to this day)
4. DMM and/or test light - Get to know how every function of a DMM works, and a test light for quick lazy checks)
5. Pliers/Cutters - Made due for years with a cheap set (and still have a couple), wire strippers are a must.
6. Screwdrivers - As said above, a bit set and driver will cover most everything later on, but starting out screwdrivers are extremely versatile tools (prying, lining up holes, scraping, used a a punch in a cinch).
7. Torque Wrench - 1/2" to start, 3/8" depending on what you're working on. Can't guess on how tight that head gasket is being clamped down (and makes sure those oil drain plugs stay in).
8. Punches/chisels - Amazing how often you'll use them.
9. Razor blades - Cleaning gasket surfaces
10. Picks - Another tool I got by without for 5 years, then once I had a set didn't know how I lived without them so long.
11. Pry Bars - I spent years using a couple of standard crowbars (still do), but even a set of $20 Craftsman bars is a major improvement.
12. Filter pliers/wrenches - If you're starting out in a shop, you'll be changing a lot of oil.
13. Flashlight - Couldn't do my job without my Streamlight, I take it with me everywhere.
14. Magnetic Pickup - Because you're going to drop ****.
15. Service cart - If you're just starting out in a shop environment, you don't want to be loading in a mostly empty 56" box and crowding precious shop space.
Then things get out of control with air and cordless tools, ratcheting wrenches, stubby wrenches, ratcheting stubby wrenches, specialty tools that you'll use once every 5 years, etc. GJ is dangerous like that.
