Anyone here know woodworking?

I'm getting a little confused. I've been handed a lot of info. So to do the half lap by hand, minimally I just need a crosscut saw and a good set of chisels?
Basically yes.
If its all hand tools then its:
1. A sturdy bench
2. A bench hook, although other holding devices will work, such as a pair bench dogs.
3. Cross cut saw.
4. A chisel meant to be struck,
5. A mallet.

But the specific choices will depends on what you will be making and the wood you are using.
Using a fine chisel on construction lumber, say stud grade Hem-fir or SPF is just going to be frustrating. Even on a better grade Douglas Fir its going be rough carpentry and chisel and saw to match.

For rougher work, a half decent 'bench chisel' is good. For finer work, and especially finish work, I like something with a longer blade for better control. At that point I'm often not using a mallet but paring.

The really fine woodworking hand tools, bought new, are expensive. Narex had a good rep as a decent but lower priced chisel manufacturer as of a few years ago. At some point you'll probably get into at least some basic sharpening. New or used chisels almost always need some additional sharpening. Hand saws sometimes can be purchased already sharpened - that's a big time saver.