Hello, Welcome to the forum!
Just a few starter tips. The best way to keep a betta is in a low-flow filtered tank, especially if you decide to cycle it. some people do, others (myself included) do not. 2.5 gallon is enough to hold him happily. Some will say a minimum of 5 gallon, but i personally believe that a 2.5 is fine.
I would recommend giving him a 30% water change twice a week if you are that committed. other wise a 50% once a week. THe gravel vac is a good idea!
Any way you can get use a closer pic of his fins so we can let you know?
Fin rot isn't really difficult to get rid of, (mine just went through it), and his fins never totally went away, since i caught it really early on. you just use this anti-biotic called "maracyn." you put it in his water for about 9 days and viola! no more fin rot
then just keep his tank really clean. also, if you are worried about the fin rot, make sure to keep his water really clean, and when you think he has it, change the water every other day. the clean water will battle the fin rot.
I have also learned, if you keep his tank warm *hint hint... Heater!* his colors will show much better. And don't worry, his color may change a little just from being in better water conditions and happier. Most bettas color will change. some drastically some not so much. But I have never seen a betta get uglier
Good luck with him! let us know if you have any more questions!