New to climbing. What do I buy?

Used shoes are great for the first pair, that was what I did too. Great choice!

For ATC & locker (which you don't need for bouldering btw), just get the basics:

ATC -
rei.com/product/812083/blac…

Locker -
rei.com/product/722353/blac…

or the smaller & cheaper:

rei.com/product/876548/blac…

You don't need chalk to learn to climb, there's no reason you have to have it if you are trying to stay on a budget.

P.S. If those nuts are from the 80's, they probably aren't wired nuts. Consider getting new cord for them. As with rope and webbing, 30 year old cord simply should not be trusted. In any case new cord is pretty cheap, REI sells it for a couple dimes per foot.