I have had 1 cat for 10 years . . . for the past 5 years we've had 3.
In terms of work, you don't have to walk a cat like a dog. You do have to feed that cat at least once a day, and put fresh water out. You also have to clean a litter box daily / multiple times a week (though for ~200 you can get an automated self-cleaning box at PetSmart/PetCo). After the first year, you should take the cat to the vet for a checkup. In terms of maintenance, that's it.
If you go out of town, cats do much better than dogs. You can have a friend come by every day or two to leave food, etc., and the cat will be fine. Dogs pine.
In terms of shedding . . . it really depends on the breed of cat.
--Long-haired (Persians)--usually these are pure-bred--have to be brushed daily (or their fur mats) and they shed a lot.
--"Medium hair" (they look long, but their fur doesn't mat)--Maine coons, for example. They love brushing but don't require it. They do, however, shed. I love my Maine--well-behaved, classy cat--but his undercoat is like fine pieces of cotton ball and gets everywhere. I had no idea when I adopted him--he looked really fluffy.
--Short-hair. These guys have the really short fur, and they shed minimally (most of the year no, but in spring, some). This is what you want.
To know what you're getting, either take a knowledgeable friend with you, or get a cat breed book and look through it before you do. I would recommend a cat breed book anyway. Even though I believe in getting shelter cats, understanding the breeds can help you as you're looking at them (to see the longer fur, etc.) and can help you understand them later. For example, Maine coons typically bond with 1 person closely, plus a general bond with the rest of the family. There are other breeds that tend to bond closer with kids. Some breeds prefer to live in groups, others prefer to be the lone kitty.
An advantage to adopting a shelter cat (or one from a rescue group) is they tend to be more loving and less prima donna-ish. Some of the rescue groups foster them for a while, so you can find out a cat's personality (and if it likes / is used to / can tolerate kids).
One important consideration is who you want the cat to bond with. If you want the cat to look up at you as the Queen of the pack, source of all good things, etc., get a male. If you don't want it bonding to you as much, try a female.
In terms of training, yes, cats can be trained. You can do reward training with a clicker (like dogs). You can do aversion training with putting noisy/sticky things where you don't want them to go--or do what I did, and just involuntarily scream when they try to climb your legs with sharp claws (now none of my cats ever use their claws on anyone indoors). Of the 3 cats I have, the best behaved is the one I trained from a kitten when I was home all the time. The one who was a kitten when I worked will misbehave when I'm gone now. The one we adopted when he was older, has been a bit slower to adapt our rules.
In terms of keeping them out of a certain room . . . if it has a door, yes. Just keep them out. If it has no door, I don't know how you'd keep them out.