The idea of exercise, as I long understood it, is to exert the muscles enough to cause very slight localized damage, which the body then repairs to be slightly stronger than the original material. I've since heard a theory that it's making the muscles anaerobic that causes the growth.
Going two or three times a week is probably best for most people; this gives muscles time to recover / repair between workouts, while not being spread out too much to have effect.
My pattern of exercise, gleaned from various sources (and possibly from my imagination in places) is to do a couple (or more) of circuits of the machines, doing a ``pyramid'' each time at each machine, that is, a decreasing number of repetitions against increasing resistance. So, for example, on one machine I might do 12 reps with 3 weights, 10 with 4, and 8 with 5 the first time round, then 12 with 5, 10 with 6 and 8 with 7 the next time round, perhaps then pushing it up until I get to just 2 reps, for strength training as I'll be well warmed up by then.
For home exercises, I recommend a set of weights, and a very simple spring-based exercise machine called a Bullworker that can be used for a variety of exercises.