Thanks again everyone for the detailed responses. I've started doing some one-leg drills on the stationary trainer at home and am slowly building up that sort of thing.

In the meantime, I spoke with the people who manage the gym I go to and told them the issues I was having with the personal trainer I had been working with. They were very cool about the whole thing and said they would be willing to let me do sample workouts with the different trainers at no charge, so I could find the one that worked for me. I'm now working with a different trainer who is a much better match. He's a cyclist himself, so he understands the theory behind building a strong base. His workouts are hard, but better balanced in that he does a wider variety of high intensity exercises, not all the same thing (squats, legs, ad naseum) like the other guy was doing. I'm tired after a workout and sore for maybe 1-2 days, but it doesn't last much longer than that.

He said he would also be willing to write up a plan for what to do in the gym the rest of the week to build a good base until the roads are good enough to start riding outside.