I did that kind of trip last year. I took Amtrak to Denver and rented a car from there.
It's a fairly significant drive to Chama (or Durango) from Denver, but I stayed overnight in Colorado Springs with a friend before continuing on to Chama next day. On the return trip from Durango I returned to the Springs.
While out there I went to C&T, D&S, the Loop, CRRM, the cog. Also managed to get back using the northern end of the "narrow gauge circle" so saw some neat stuff.
Amtrak westbound gets into Denver around 9 AM. Denver's station (terminal?) has a phone with free calls to call the rental agency to have them pick you up. I rented from Hertz, I suspect in a year or so you can take RTD (light rail) from the station to the rental place.
Amtrak eastbound gets into Denver around 9 PM. Had to return the car early so ended up taking RTD around and stopped off to visit Caboose Hobbies (wow what a place).