equipment:

It seems to me that if you have to ask, a better place to begin is developing a solid knowledge of what climbing actually is and, from that, determining your particular level of commitment to the aforementioned. Once you know what you're doing, you'll know what you need.

Sport climbing and bouldering are relatively inexpensive and widely accessible , however trad climbing, aiding, big-walling, ice-climbing, and alpinism are all quite financially exhausting. You could spend a small (or large) fortune pursuing any of these. What's more important is having a solid knowledge of what it is you're doing, so focus on that first.

I'll let your capable and experienced partner give you that list.