I live in a two year old house built on a slab. There is a faint sound of running water in the bathtub drain. This is continuous even in the middle of the night. When I flush the toilet there is a "gurgling" sound in the bathtub drain. What's going on, and how can I fix it?

You may have several issues here. The gurgle sound you hear is the system attempting to vent through the tubs trap. This tells me that you have a blocked vent, The solution would be to snake the lavatory roof vent. Run out enough snake to reach the base and 15 feet more.
As for the sound of running water you may have a hidden leak.
To check to see if you have a hidden leak first shut down everything in the house, no flush, no drinks of water, and if you are on a meter go out and check the little pointer in the gage. It should not move or creep. If it does you have a leak. If you're on a pump then check the pressure gage after the pump builds up to pressure and shuts off. The gage should not fall and the pump come back on. If so then you have a leak. Hope this helps and thank you for rating my reply, Tom