Welcome,
Get out on your bike more! I built my fitness up fast last year just by riding on the road. You can be good on an excersice bike and a turbo, but when you get on the road you just wont be as fast.
Join a club. This is what i did. You will end up doing more miles, which will get you fitter.
I dont bother with riding on a turbo or excersice bike, i just go out, even if it rains. It will just make me harder.
Also, you say you went up a hill that fast, what are you classing as a hill?
Whats your times to do 10miles, 20miles, what do you normally do in an hour.
As for resting heart rate, mine is normally around the mid 50bpm. Its been down lower before when ive been really relaxed.
Get talking to older members in your club. Someone i know helped me get better at riding. He used to race, and be pretty good at it. Well be riding along and he will be moving in on me, touching shoulders and sometimes even making his front wheel touch my backwheel. Its to get me used to stuff for racing, and its a good laugh