Q: Transferring data with new service provider and SIM card

I have found possible steps in Apple Support to transfer info from one carrier to the next as I switch SIM cards, but I'm not positive as they don't exactly outline my specific situation: 

I have an iPhone 4s, Model MC923LL/A, utilizing OS 8.2 (12D508) currently on an AT&T network. I am switching to a different provider and have received the new SIM card. My question is this:

How do I successfully switch SIM cards and not lose any data?

I'm not exactly clear on how SIM cards work, so am not sure what is stored in the phone, and what is stored on the SIM. It would be really great if I didn't have to re-enter all of my passwords for various apps and web sites, and I am concerned about losing any contacts.  I am pretty sure I'll be losing my saved voicemail, but if that can be helped, I'm all ears.

Thanks in advance for any help.