I vote yes on trying Spinning classes to see if you like them. I teach Spinning at my gym year round. In the winter, a lot of roadies come to the gym just for that. For a lot it's more motivating than getting on their indoor trainers alone. Plus, you might find that you make some new outdoor cycling friends to discuss the sport with while couped up inside during bad weather.

Storage... lol, my stuff is all over the place atm. I was thinking what about clear plastic under the bed storage boxes? Jerseys, shorts, etc. could be rolled and placed inside. Sort stuff out by category and label the boxes on the outside. That way if you want something, it's not such a mess to dig through stuff to find it.

I second that you just need a wicking tank top for indoor training. You are not traveling a distance around that you need the pockets of a jersey. Sports bra. Maybe a pair of shorts or two for the trainer. Or, if inside you could use a gel pad seat cover (in place of a chamois) and wear whatever for shorts for spin class.