You'll need the right supplies and also a good amount of knowledge on bettas. Most 'rescues' have things like fin rot, so if you are not aware how to treat these things, you should do research before doing any rescues as you can just make the condition worse.
Things you'll need are a nice sized tank, a heater, plenty of plants and AQ salt (for fin rot, velvet, etc..) and Epsom salt for swim bladder. I would assume that most pet stores don't overfeed their fish, so the chances of you rescuing one with swim bladder is probably slim, but it's best to have it around just in case.
I think it depends on your defition of a rescue. Some think purchasing a Betta (sick or not) from a store is rescuing & others not. Some stores will let you take an ill Betta for free which I guess is rescuing it from almost certain death. Some get ill or injured Bettas from people they know or someone advertising one they want to get rid of. There are probably other ways but I think those are the most common.
Yeah, you go to your LFS, stick one in your sleeve, and it's 'rescued' from the evil prison, joking. What do you mean 'get involved?' Once you buy one, you will have been infected by the betta-virus. That's how you get involve. Just pick the one you like, but keep in mind the more active they are at your LFS, the more likely you will not have to treat him/her and your life will be much easier. As for rescuing, you just take him/her home and give them a better life. That in most cases, would mean good food, water, home. Good luck and don't hold back on the questions!
The tank needs a large surface area. They breathe air directly from the surface, so the more surface area they have, the better. You will probably find that they spend a lot of time at the top of the tank.. So having plenty of plants that reach the surface is ideal because they rest on top of them. Do not have so many plants that it completely covers the surface area because you can drown your fish..
Ideally, it should also be at least 2.5 gallons or larger.
For decor, real plants or silk plants are best. Plastic plants can sometimes be sharp and hurt your bettas fins. Some plastic plants are fine if you make sure that they are not sharp. If they snag on pantyhose, they can snag your bettas fins. For caves or other decorations, same thing. Make sure there are no sharp parts. Also, make sure there are no holes that you cannot fit your thumb into. They can get stuck in small holes.
For filters (if you decide to get one), just make sure it has a low flow. Bettas long fins make it hard to get around in fast moving water. Try to make the water as still as possible.
As for heaters, get one that is adjustable. The preset ones are often inaccurate and are not dependable. 25 watts is usually good for 2-7 gallon tanks.
For food, buy pellets. The flakes can get stuck in their fins and its hard to feed a set amount as they are all different sizes. Personally, I use Aqueon pellets. They are small and easy for my bettas to eat. Omega One and New Life Spectrum are good brands and high in protein.
Make sure you have a water conditioner, and some AQ salt for treating potential illnesses. Epsom salt is good if your fish develops swim bladder from over feeding.
Test kits are also good if you plan on cycling a tank or if your fish start showing signs of sickness. With a test kit, its easier to pinpoint if it is high ammonia/nitrates/nitrites that may be causing a change in a fish's behavior. The API Master Test Kit is the one most often recommend here and is slightly expensive but also includes over 800 tests.
If you are tight on money, the most basic items you need is a tank, heater, food, age/dechlorinated water, and your betta. It's as simple as that. If you keep up with PWC weekly you should be fine. All other supplies are optional, but I will add that IAL is recommended. I would also recommend NLS pellets, thera A. For dechlorinator, I use prime, but back then, I would just let my water sit for 2 days prior to using and fish were okay. I use catalina titanium heaters and eheim jagers, they work fine like many other heaters out there, but make sure you read reviews first as not all heaters do what they are said to.