Every wireless router will include an option to apply encryption -- the most compatible and secure at the moment is WPA wireless security.
I'd read a few router reviews to see which score best for signal quality -- more than the technology, your signal will be affected by your environment, potential interference, choice of channel and position of equipment. See setup instructions which come with router.
Frankly wireless is not very good for gaming so I would also have a look at mains plug adapter networking solutions -- though a simple ethernet cable is faster and much cheaper.
I live in a apartment so I dont care about the range, just the signal strength. It will be used for gaming and basic home computing. Ive never had one, do I have to worry about my neighbors using my signal??