Can I purchase my own modem?

You can place a compatible modem BEHIND the AT&T router but you cannot replace the AT&T provided router.

so basically you can connect anything you want to the back of the gateway, but it will not get you faster speed. at most it will help with the wireless range and maybe if already have multiple devices connected to a pre-existing router. your internet speed is capped by the transportation type and distance to the pfp/cross box. 

att technicians are only responsible for service up to the gateway, which means they can chose and encouraged to charge you regardless if they can fix the issue if it's a customer equipment. 

at the end, do whatever you want man. don't be surprised if it does not improve performance and don't call att unless the provided gateway is not working properly.