I'm not really knowledgable about fin rot and the like, but aquarium salt and clean water will be a huge help. You can also buy some Bettafix.
It's a good thing you're getting him a bigger tank, I bet he'll love it. ;)
As for tanks, this
is what I'm planning to buy if I want another betta. Normally kits come with a lid with a light, a filter, water conditioner, food (which is normally tropical flakes) and sometimes a plant or two. Normally they don't include a heater, so if you don't live in an area where it's constantly 76-80 degrees you'll have to buy one.
I don't know anything about snails or ADFs, sorry. You certainly won't be able to fit another fish in. Maybe a ghost shrimp.
Cleaning isn't as hard as it looks. First you should wash the aquarium with warm water to get rid of any dust or residue. Later on when you add your fish and such, you should do a 20% water change each week if your tank is cycled. if it's not, you'll need to clean it much more regularly.
I don't know if they make gravel cleaners small enough for a 2.5 gallon tank, but you should look around and see if they do. While you're changing the water you should also clean the gravel to get rid of any fish waste or left over food that's fallen to the bottom. Oh! Forgot to mention that when you first get your gravel, you'll need to wash it well in warm water to get rid of any dust/residue.
As for plants, silk are far more easier to maintain, since they're not alive. :P
Real ones require light and fertilizer, but the pros are that it adds a nice natural look to your tank, helps prevent algae, and helps with the cycling of your tank.
Which brings us to your net question...
Cycling is a good idea because it stabilizes your tank, and you'll end up cleaning and changing the water less often. You'll need a filter (like I mentioned, usually comes in a kit). To start the cycle, normally you'll need to add the fish to introduce ammonia, unless you're doing a fishless cycle. that's adding amonia manually, but I don't know anything about that.
What happens is that your betta will start producing ammonia in the form of any left over food or waste. A lot of it will be deadly for your fish, so you'll need to do frequent small water changes at first. The ammonia will start to increase slowly, causing a 'spike'. The filter starts to then produce beneficial bacteria that convert ammonia to nitrite. Nitrite is also harmful to your fish. That'll eventually spike too, then other bacteria will form that convert nitrite into nitrate. Nitrate isn't very harmful to your fish unless it's in large amounts, which is why you do weekly water changes.
It's very helpful because you can leave your tank for a few days knowing ammonia levels won't rise and the like. Also much less cleaning for you.
Usually cycling takes about a month, I believe. You can speed up the process by buying live plants, or adding something from a cycled tank (filter media, water, gravel, etc) to help speed up the growth of bacteria.
I did both and now my tank is almost finished cycling in two weeks. :)
Anyway, it's not completely necessary to cycle your tank. You CAN treat the water and put your fish in. But again, in the end it's more cleaning for you.
Phew! That was a long post. Hope I didn't forget anything. :P