What can you do with a MAC address?

If I gave someone my MAC address so they could provide me wireless internet, can they do anything else with that? Can they infringe upon my privacy or access any personal information on my computer?

No,

They can infringe on your privacy a lot more just by having you go through their network. 

Oh, I was assuming you were asking what types of neat things you can do with MAC Addresses.
I usually just change my MAC Address whenever I want a new IP from Citywest. Just gotta provide them with my Username/Password