Is it the phone or is it your signal? If it's your signal, try disabling 3G when you are home. You may have more bars on Edge and a more consistent signal. The data speeds won't matter if you have WiFi at home. The only drawback is the Edge speaker interference. Regarding the phone shutting itself off, restore it to factory settings. If it continues to shut itself off, tell them you restored it and it continues to shut itself off/spontaneously reboot. Always restore to factory settings before going to an Apple Store for something like this. It may save you a trip.
Apple store would have been what I would have said. I saw this earlier and was waiting for other comments first. There was that issue with one of my phones as well and there are a few people out there with that problem. Now that said. Nobody has been mean! They tried to help with the little information you had given.
Is there a problem updating the iOS ?
If you are only having trouble at your house get a network extender, its is a little pricey upfront (~$100 or $250 depending on which model you go with). But it fixes it and there's no monthly cost after that. If you decide to get it from Verizon, mention that your service is crap at home. I got them to give it to me for half off ($125). Here's some more info: http://www.thebestiPhone.com/?p=94