Bathing, if you're looking at just removing sweat after a hard ride or getting a little bit of the surface dirt off, you can just use a hose and squirt the horse down, using your fingers to rub where the water is to loosen and wash away some of the dirt.
If we're talking a full bath, get yourself some horse shampoo and conditioner (there are some brands of human shampoo/conditioner that work really well on horses too, but you do need to be careful because some brands can cause irritation), a large sponge (car washing sponge is great), an old body brush, a sweat scraper and some old towels.
I prefer to leave as much of the horse dry for as long as possible. So I start by washing the tail, rinsing it then adding conditioner. While the conditioner soaks in, I'll start washing the mane and gradually work my way back to the rump, wetting with the hose and massaging the shampoo in with the sponge and body brush depending on how dirty the horse is. Once you have worked the shampoo all the way through, rinse it off thoroughly - this is very important, you cannot leave soap in the coat because again it will cause irritation.
When the water running off is clear, use the sweat scraper to scrape off the excess water, and rub down with towels to draw out the last of the moisture.
Combing the mane and tail - I use a metal detangling comb, then work through with a wide bristle hair brush. Most horses are happy for your to work the tangles out of their mane and tail, just watch that you don't stand directly behind the horse while brushing it's tail. I always start at the bottom of the hair, and detangle sections at a time. Be careful brushing the tail - tail hairs grow VERY slowly and it is easy to pull out hairs, thinning the tail. I very rarely brush a tail out, only when I am competing.
Trimming the mane, I pull the mane as taking to it with scissors can look pretty horrendous. But you can clip/trim the bridle path and wither to keep hair out of the way of tack. The tail, I level the bottom off just above the fetlocks, and pull the top (sides). If you're not competing, I wouldn't bother with pulling the tail, and just level it off before it starts dragging on the ground.