If you get a normal Arduino (there's a bajillion variants, so I won't address all of those) you need any computer made in the past 15 years running Linux / Windows / Mac, and a usb -> usb A connector. Most Arduinos include that. I suggest, however, you buy some electronic parts - a couple of LEDs or sensors - via a kit and a jumper kit (a bunch of pre-bent color coded sized wires) and a breadboard to experiment.

Also, check out for local hackerspacers, as there will definitely be people that to set you on the right path.