I have both kindle and I pad. I have Kindle in the nursery for frequent feeds. If you're wanting it more for the Internet than reading books then kindle isn't the best option.
I find the Kindle a little frustrating to surf on (even mn). Don't know about game playing as I just occasionally play word games or sudoku on it, his needs and expectations would be higher than mine. Maybe you could research other tablets that are cheaper than I pads (can't help there I'm afraid as am useless with gadgets I have never mind those I don't) You can put a kindle app on most things and they work brilliantly.
I would like a kindle for myself and maybe one for DS to encourage him to read. With the fire you can surf and play games etc? Would he actually read on it or just play games. Not going to stretch to an I-pad but what is the difference. Is the paper white best for just reading books on? All very confusing! Any help or advice would be great x