Do I need to remove the sim card from my old phone and install it in my new phone

That depends.

If you had your cellular service activated on the new one by your carrier either with a new sim card or an eSim then no.

If you have not had your service moved over, then yes, you would need to move your Sim card over to get your service and number on your new iPhone.

After I move the sim card from my old phone to my new phone, what do I do with the extra (new) sim card? Put it in my old phone?

You can if you want to, or keep it in a drawer in case you ever want to use it.