Definitely either the Samsung Galaxy S III, or the Samsung Galaxy Nexus.
Both phones will run almost any game thrown at it, at blazing speeds. If you want a faster phone, get the Galaxy S III. If you are more for software and hardware appearance or hardware form factor, get the Galaxy Nexus. Both have large screens, and feel great in your hand.
The Galaxy S III is quite a bit more expensive, not sure if you're looking for a phone on-contract or unlocked. Unlocked the Galaxy S III is around $600, and the Galaxy Nexus is $350.