Previous posters are correct on how to remove the cassette and that Shimano and SRAM will be exactly the same. You can buy the splined lockring remover and chain whip (or make your chain whip fairly easily) at any local bike shop.
Front chainrings do last longer than a chain and cassette under most riding conditions. Inspect them, if they look like shark teeth then they need to be replaced.
The rear disc will mount to the hub one of two ways. If it's a Shimano disc, it may have a splined lockring, same as the one for the cassette. Some Shimano and any other manufacturers will use a 6 bolt pattern. This is more prevalent, and the bolts are T25. If the rotor is really bent, pull it, wrap it a paper tower, lay it on a perfectly flat surface, and whack it all over with a non-metal hammer. Disclaimer: Be careful here, you can do more damage than good.
Safer (and preferential if it's only a little bent) is to simply bend it straight while on the bike. There is a nifty tool for this, it's sort of a two pronged fork, and the cutout fits the rotor perfectly. Use the caliper as a reference, and straighten accordingly. You can make your own tool with a small, cleaned adjustable blade wrench.
If you replace the shifters, the cables themselves will likely come with shifters. If your housing is in good shape you can probably reuse it, just lube it prior to inserting the cables. Tri-flow, or any other teflon base chain lube will work. Just dribble a few drops in the housing prior to inserting the cable.
FWIW, I prefer Shimano shifters and deraillers to SRAM stuff. They just seem to shift a little quicker and smoother. If your chain and cassette are pretty worn, you're going to see vast improvements in shifting by just replacing those, so that should fix that issue. Also, even if you don't replace shifters or cables, still lube the cable/housing periodically.
SRAM is good stuff, I just think the Shimano stuff generally works a little better. I myself run a SRAM cassette and Shimano shifters/derailleurs on all my bikes.
Feel free to ask more questions.
BTW, I'm employed as mechanic/tech (whatever we are called these days) at a bike shop, been doing it for 10+ years now.