SMS and MMS logs, while not transferrable, are in theory rescuable, depending on what kind of smartphone you own, but it requires a lot of legwork on your end. You won't be able to add them to your new iPhone, however; you'll simply be saving them to your computer. There are a variety of different programs available for exporting messages from your smartphones--SMS Backup & Restore for Android appears to be one popular option. As I haven't used it, I can't personally recommend it, but you can always search Google to bring up more options.
If you're upgrading from a feature phone
Yes, it's finally time to toss that Razr aside for something a little more full-featured. But what about your contacts and your photos? If you're on a phone with a nano-SIM card, it's easy enough to rescue the first; for the second, you'll need Bluetooth support or a connection cable for your device.
You can transfer your contacts one of two ways. If your old device uses a nano-SIM card, you can copy all your contacts to its SIM; once you've set up your iPhone 5, you can copy those contacts by swapping out your iPhone's card with your old nano-SIM and heading to Settings > Mail, Contacts, Calendars > Import SIM Contacts. Once the import has completed, eject your old SIM and put the one that came with the iPhone 5 back in its tray. (If you're worried about doing this yourself, you can always head down to your nearest Apple Store--any employee should be able to do it for you.)
If your phone has a SIM card that doesn't fit in the iPhone 5, but it comes with a USB cable for connecting it to your computer, you may be able to transfer your contacts (and your photos) by exporting them through software. (This, obviously, depends on your phone; check its manual or do a Google search for specific transfer information.)
Activate your iPhone 5
If you purchased your iPhone in the Apple Store, you may have already activated it with a Specialist by your side. But if you've received your new gadget in the mail, or you opted not to activate immediately, here's a guide on how to do so.