I used to do A LOT of weight-training for soccer and track/cross-country running. However, with my switch to bike-racing, I only did 6-weeks of weight-training in the winter. Surprizingly, I actually double my strength in the winter to be even stronger than when I was playing soccer or running AND I don't build any mass and am trimmer and leaner than before.
That's helped me tremendously on the bike as my legs are operating at a lower percentage of their max for any given effort in Feb. as compared to the same ride in Nov. (muscle-strength naturally goes down as the race-season progresses).
Gym workouts also helped my back a tonne as well. My first month on the bike, I was gung-ho and rode way too much. My back would be sore and ache for hours. It got worse and worse each time I rode as well. The doctors' advice was bed-rest (which was about all I could do at that point anyway), take a lot of pain-killers and stop riding, which was NOT acceptable. Going to a PT and chiropractor was much more helpful. They put me on a training program on the gym to strengthen my back before I could get back on the bike. Started out with basic stomach crunches and back-lifts. After a month I was tripling and quadrupling the back-lifts with putting more and more weights on. Got back on the bike after that and WOW, no more back-pain, ever! Did 10-years of bike-racing and never had a problem with my back again. Might have overbuilt my back and upper-body though as when I got into track-racing, I easily ripped a couple of handlebars and stems in half in a sprint...