4 years ago I was in a similar quandry. I joined and gym and started seriously training to lose weight at the same time I was trying to go from struggling to ride a mile a day to becoming a serious road biker.
I was lucky enough to have a trainer who knew enough about biking to understand that what I needed was core strength, arm and shoulder improvement and a lot of pylometrics and alternating upper and lower body sets to increase stamina. He also understood that I didn't want to build bulk but decrease body fat and increase lean muscle, both for weight loss and for overal fitness.
I worked with him for a year and a half. He was followed by a former Marine with sports training certification who basically believed in hitting every muscle every time for sustained strength. That meant that we did a lot of reps at light weight with a lot of pauses and holds.
My current trainer is a runner but in the interim I have learned enough about my own physiology to be able to realistically advise him on some of the stuff that I need or find beneficial. He is ASM certified, a trained kinestheologist and has enough psychological training to not listen to my whining butt understand when the old bones and arthritis are hurting.
I know enough about my own mental state to understand that this has to be a long term committment since I will not push myself hard enough, nor ride hard enough without some other discipline. Having said that, I would agree that the gym days are definitely for core and specialized muscle strength and cardio, whereas the bike training is for speed, intervals and endurance.
Training is about the only thing I spend money on but it has worked for me to the tune of 125 pounds, going fromm a 47% body fat to 25%, dress size 2xl to 10 petite and from struggling to ride a mile a day to routinely riding 40-60 miles 3 x week and training 3 hours at a stretch 2-3 times a week.
but that's just me and my 2 cents worh
marni