It sounds as if it is a serious software problem on the phone itself rather than a hardware issue or a carrier service matter.
Probably the best thing to do would be to connect to iTunes and do a restore. Assuming that completes successfully, start by testing basic functionality such as SMS, and email. If they work properly, then test the rest of your apps and seem if you encounter a problem with any of them.
If on the other hand, the restore does not fix or at least help identify the problem, there are really two choices:
1) do a device firmware update (DFU) to install a clean version of the operating system and manually restore data and apps which would avoid replicating any corruption in an existing backup
2) returning the device since it is within the return period. The problem with that approach, apart from the time/effort involved is that if the problem is corruption in a backup you will restore to the new phone or incompatibility of some apps you are running with the iOS version you have, the problem will likely recur. Thus, it might be better to go with 1) first. That way at least you should know what the issue is and how best to proceed.