If you're planning on waxing your underarms, go to a professional and have a hot wax treatment as they're great for smaller areas and hurt less in my opinion. The skin there is incredibly sensitive, and unless the hairs all grow in one direction, then application is going to be a b***h. As for your legs, that's a lot easier.
Make sure the area is nice and clean (no oils or body cream) and try and use a
pre-wax treatment to prepare the area. Look at the direction of hair growth, you want to apply in the same direction the hair is growing, not against and try to do this in one thin application. Apply your strip and rub the area firmly, before removing the strip as quickly as possible in the opposite direction that you applied the wax.
Paper strips are a lot easier to work with in my opinion and they leave less fluffy mess.
If you're worried about removing your own skin, test the temperature of the wax on your wrist first to make sure it's no going to scald you, and apply a
very light layer of talcum powder over the area you're waxing after you've applied your pre-wax treatment. As for bruising, just be careful during your ripping motion. You just need to be quick, not violent. Unfortunately some people do just bruise like a peach when it comes to waxing.
Invest in some proper
after wax lotion as well, this will help sooth the redness and 'chicken skin' effect a lot faster and help to prevent any nasty little infections. No hot showers or baths for around 24 hours afterwards and no fake/spray tans.
As for ingrown hairs, just use a really good exfoliator when you're in the shower, preferably a
sugar scrub that also helps moisturize the skin.
Well, that's my brain picked. Good luck!