As long as you get an unlocked cell phone, all you have to do when changing phones is insert your SIM card into the phone and you're good to go. If the phone is locked, you will have to call the service provider to get the unlock code. Your SIM card will carry over all text messages, recent calls and phone book contacts that you have stored on it. Your phone number will still be the same, but if you want to change that too, you can call your service provider and request a number change, which will usually cost a small fee.
Make sure your new cell phone is unlocked, especially if you buy from a private dealer, like Ebay or Craigslist. If it isn't, your SIM card won't work.
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Take off the battery cover of the new phone and remove the battery. Slip or snap the SIM card out of the new phone.
Remove the SIM card from your old phone and insert it into your new phone using the same method you used to remove it.
Replace the battery and turn on your phone. Any contacts saved to the SIM card will still appear in your phone book. If you took a snapshot of each contact to go along with their number, the pictures will not carry over.
Set up your phone's settings to your liking. Depending on your phone, this could take a while. On the other hand, you may only have the option to set the volume on your key tones and select your ringtone.
Select a background and/or theme for your phone.
Download any applications or programs that you want to download to your phone. Downloads are also stored to your phone's memory, so they will not carry over from your old phone.