Hi!
Start with a weight that you can lift for 8-12 reps using good form. If you can do more than 12 reps, go up in weight. If you can't get 8 without "cheating," i.e., breaking form, go down in weight. Do 3 sets of 8-12 repetitions; if all three sets seem about equally easy, then odds are pretty good that you could go up in weight.
The dumbbells you use will vary depending on the exercise and the muscles you're using, so it will require some experimentation. Take along a little notebook and write down your lifts so that you can remember where to start next time.
The good form is key: e.g., don't turn a biceps curl into some ridiculous full-back-extension-swinging-grunt thing like you see so many people (usually college-aged men) at the gym doing. (If you're at a gym.) If it means using a lighter weight than everyone else to start out with, so be it.
As for how often to go up in weight, it really depends on the weight at which you start, which body part and exercise, what the jump in weight is to the next-heaviest dumbbell, how often you train, how much protein you're eating, how much sleep you're getting and/or stress you're under, and so forth. "Newbies" generally increase lifts faster than people who have been training for a while, in part because their bodies are still learning the new motions, but it is still going to be highly variable.
Don't be embarrassed about what you lift! First of all, no one at the gym really cares: they're concentrating on their own workouts, not yours. Second, everyone starts somewhere, and you'll get stronger if you stick with it. And third, keep in mind that you paid the same amount for your membership as Jane BodyBuilder (well, unless she's getting a discount under a sponsorship deal), and hence you have as much right to the gym space.
Kim
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