I bought a Lux TX9000TS programmable thermostat to replace my Carrier non programmable thermostat. The Carrier has a S and S1 terminal, which I think is for an outside sensor, which I have never seen outside my home. Can I just cap these off and not use them, and connect the new thermostat, or do I have to buy a thermostat with the S and S1 terminals?

The carrier may have been using the outside sensor to lock out the heat pump(if that's what you have) or the homeowner may have just been using it as a outside temp reading. You need to figure why that was connected originally.