If you are going with a different carrier, then there should be no need to give them the other phone for anything. You can do with it what you wish unless you have to return it for some reason.
SIMs are just for service. Unless you save say contacts to it, which is about all some will allow for. If they allow it. SD cards are for files, data, personal stuff, etc. However, if you are getting rid of the other phone or giving it away, then remove the SIM and destroy it.
When it comes to personal info, that you can easily save to a SD card and transfer the card to a new device (if they use the same type of card) or upload the files, contacts, etc., to a google account, or connect the phone you have to a computer (via USB cord) and save your contacts, files, music, etc. to an empty folder on the computer. Once you have the new phone you can then transfer it all to the new phone from which ever you saved it all to.
Now you can't transfer apps. So either the company you are getting the phone through (if doing in person) may be able to move the stuff to the new phone, maybe.. big maybe) otherwise you will need to redownload all those apps to the new phone from google play.
Be sure you wipe all your personal stuff, everything, from the old phone. It would be wise to do a full "Hard Reset" if you are not keeping it, turning it in, or giving it away. Instructions for that are below.
NOTE: Doing this will clear 'all' the data on your phone, so it is suggested you not choose this option unless you have to, or aren't worried about losing anything on the device.
Be sure you back up any data you want to keep, to a computer or cloud account, before doing the reset.
To "Hard Reset" the phone...
1. Make sure that the phone is actually powered off.
2. Remove your SD card, if there is one in the device.
3. Press the "Power" and the "Volume Up" buttons, together, until you see the start-up logo on the screen.
4. Once you can see the logo, release the buttons and allow the device to enter the "System Recovery Mode".
5. Next use "Volume Down" button to get to the "wipe data/factory reset" option.
6. When in recovery mode the "Volume Up" and "Volume Down" buttons are how to can navigate and highlight what option you need.
7. Once you have selected the "wipe data/factory reset", press the "Power" button to select that option.
8. On the following screen, repeatedly press the "Volume Down" button until the "Yes - delete all user data" option is what is selected.
9. Then press the "Power" button to hard reset the phone.
10. If you took your SD card out in step 2, when the device is up and running again, reinsert the SD card.