Sync from old IPhone to new IPhone

If you weren't having problems with your old one you can restore the new one from a backup of the old one. Connect the new iPhone to the computer which you will sync with (the same one you synced the old one with). Select restore, it will update the new iPhone and install the OS again. Once it's done it will ask if you want to set it up as a new iPhone or restore from backup. Select the most recent backup of your old iPhone. Now your new iPhone should be setup exactly as you had the old one; pics on the camera roll, texts, contacts, and all that should appear as if you never even changed phones. You will need to perform an additional sync to restore the media; music, videos, and apps.

What "old phone?" Randy's advice will work if that old phone is an iPhone. If it's not, you need other options...

If the phone has a SIM card with contacts stored on it, you can insert that SIM card into your iPhone and use Settings > Mail, Contacts, Calendar to Import SIM Contacts.

Else, you'll need to transfer the contacts from the old phone to your computer (via Bluetooth, USB, etc., if supported by that phone), then sync them to the iPhone.

Alternatively, your old carrier may be able to export the contacts and save them to a USB flash drive if you go into one of their retail stores. You could then upload them to your computer and sync them to your new iPhone.