Do I have to try to mine bitcoin in order to verify peoples transactions?
Verifying the validity of transactions is not the same as mining. Verifying transactions means that you are just running a full node; mining is taking those transactions and putting them in blocks so they are committed to the blockchain. You can run a full node and verify transactions, but you won't get compensated for it. Only miners can earn money as they are spending the energy and work required to generate blocks.