How do you switch out a sim card on a cell phone?

Okay so I asked this question but wasn't happy with the answers. My phone is new but was free when my dad got a family share plan for me and him. It just does some annoying things like turns off whenever it wants even when charged fully, it has horrible reception, it drops calls, and sometimes won't even ring...I've been told that people tried to call me and it went to voice mail but i don't have a message! It will also go straight to voice mail (sometimes) when I'm in another conversation w/ another person. Anyways...it's usable but annoying me. So I was told that you can just go get a prepaid cell (I plan on getting a really nice one...i don't want another version of my phone i have now) and you can just switch out the sim card..? Is this true? Does it have to be bought at the AT&T store since thats where my phone is from I have now? Does it screw up billing? Do you keep your #? What about your saved contacts? Someone give me DETAILS.
Forgot to mention that my dad gets the bill...I need to know that it won't screw anything up price/billing wise.