Kicked off internet even though still connected wirelessly.

I bought a new iMac about 3 weeks ago. My router went out a week later and had to be replaced. I got the new router in and the computer working wirelessly, and then I downloaded the new OS X. Ever since then I get kicked off the internet about every 2-3 hours. (The message says Safari can't open the page. Server isn't responding.) If I restart my computer I can get on the internet again. I have checked to see if I can get Wifi for my iPad and iPhone and they work fine, so it doesn't seem to be a router problem. Anyone know what I should do? (On a scale of 1-10, my computer knowledge is about a 3-4, so please keep that in mind if you can offer a possible solution.)