Starting a strength training program can be really intimidating. Before I became a trainer, I had no idea what areas of my body I should be training and how often I should train. The good news is that it's not as mysterious as it seems when you are standing in a gym staring at all of the equipment!

Strive to strength train a minimum of two, non-consecutive days. Your muscles need a day, or more, of rest in between sessions in order to repair themselves. Lifting weights damages muscle fibers (creates microscopic tears in the muscles you’ve worked), so rest days give your muscles time to reconstruct, recover and increase in size.
Strength training can be achieved by using your body weight, machines, free weights or various pieces of equipment such as medicine balls, swiss balls, BOSU, TRX, bands, etc.
Strive to do at least one set of 8-12 repetitions of 8-10 exercises that target the major muscle groups: chest, back, shoulders, biceps, triceps, abdomen, quadriceps and hamstrings. Check out the images below to help identify these muscle groups.
Set up a session with a personal trainer to help you come up with exercises that will target these muscle groups. If you are planning on strength training from home, you can still contact a personal trainer to help you create a program. I've met with a lot of clients who just need help getting started.
Adjust your weights according to the "two rep" rule. If the last two reps of an exercise aren't taxing, you might need to consider going heavier. If you can do dozens of reps at the weight you are using, you need to pick up the next heavier weight. In body weight exercises, you can add additional repetitions or modifications to your strength exercises to make them more challenging. For example, instead of doing push-ups on your knees, try to do some negative push ups on your toes. Remember that you don't want to increase everything at once: sets, number of workout sessions, repetitions, weight, etc. Pick one area to increase at a time.

The American College of Sports Medicine (ACSM) has a wonderful brochure that will share more information with you.