Open Contacts. Edit a contact. The dropdown at the top (which won't drop down unless you're adding a new contact) shows you whether it's a Phone, Device (same thing), SIM or Google contact.
See the Contacts section at
Backing up an Android Device. Convert all your contacts to Google contacts, then sync. (You can then go to
Google Contacts to clear up any duplicates, etc.) Put the SIM into the new phone and sync contacts (after entering your Google account in Accounts) and you'll get back all your contacts (which is
why all contacts should be Google contacts - you can't damage or lose the site).
For texts, use
SMS Backup & Restore to back up from the S5 and restore to the 3T.
What's on the SIM card, other than contacts (which is a holdover from back when SIM cards were the only user-accessible storage on a cellphone) is the SIM's number - which it encodes and sends to the tower. The carrier's computer does all the rest of the work, like translating that to your phone number. (And you
do have to get a new card, because there are no "make it smaller" adapters [there
are "make it larger" ones], but the carriers don't charge for SIM cards [some MVNOs do, though].)