You have better control of the heat (and therefore time as well) in the oven than on the stove top which makes the the oven a bit more forgiving. Braising on the stove top requires occasionally checking and stirring to make sure nothing's sticking to the bottom of the pot, but braising in the oven does not.
While your cheap, stainless pot will work, do yourself a favor and invest in a decent braising or dutch-oven style pot when you can. It's true you don't really need one, that you can braise in just about anything that doesn't actually leak, and also true that restaurants often cook their braised meats in inexpensive pans covered with aluminum foil.
True, but... A heavy, enamel over cast-iron pot works better for the initial braise, is more forgiving (there's that word again), and will make your life easier and more pleasant.
The idea with almost any braising any meat is cooking low and slow to make the meat very tender, but not cooking so long the meat breaks up into stringy fibers. With few exceptions, e.g., ropa vieja, stringiness is to be avoided.