Well. You gotta start somewhere. Odds are, unless you've already got a tinkerer in the family, you'll be starting similar to me - with a variety handtool set of some size or another. Nowadays they have home/garage/hobbyist tool sets that would likely be an ideal place to start. That wasn't the case when I got my first tool set. I got a 300-something piece Craftsman Mechanics Set for my birthday/Christmas when I was a teenager. It got me through many repairs - some around the house but primarily on cars/motorcycles. As time went by, I slowly began expanding my collection as needs arose. One thing available today that I wish I had from the start is ratcheting wrenches. They've saved my sanity a few times now. Newer cars... unless you disassemble it all the way down to the part you're trying to fix - which takes forever - it's quite difficult to get to many parts. Especially failure-prone parts like thermostats, waterpumps, and the likes... what are the odds... any hair's width of space that you can save may turn out to be crucial.
Another one - though I'm not sure if it can be called a handtool, is a bench vise - well any vise really - but a swivel bench vise is HUGELY useful. Though most professionals would likely scoff at me, I'm using a 5" that I got from Harbor Freight for $40 after coupon. When I got it, I tore it down, completely cleaned it, ground and polished all contact surfaces as well as the little anvil, and put powdered graphite on the threads. It's been fantastic. I couldn't find a name brand vise of this same size beat to shit for this cheap. That was actually my original plan - fixing a trashed name brand one. But I have zero regrets with the direction I chose. There's a few pictures in the slideshow below.