Could it really be possible for a rider, like me (who is fat but loves cycling) to improve his skills and gain stability for riding a good mile? If Yes, How?
Yes, stick to it and keep doing it, each time will get you that much closer to your goal. It is very possible, start slow and stay dedicated.
I actually am jealous of where you are! At your stage of training, you will see enormous improvements in your fitness and athletic performance by any measure you like, in a short period of time and without much effort. I haven't seen 100% or 200% improvements in anything in a six month period in probably 20 years.
At the stage I'm at, on the other hand, I have to put in a Herculean effort to see even small performance gains. Thankfully, I bike because I like it. :)