Where to start (strength training)

I second the suggestion about a session with a personal trainer.
If there's not a reputable one available at your gym, here's where you can search for people in your area who are certified by the American College of Sports Medicine:
http://members.acsm.org/source/custom/Online_locator/OnlineLocator.cfm

I did a blog post discussing exercise, goal-settting, the different ways to lift depending on what you want out of it, etc.
http://www.myfitnesspal.com/blog/MKEgal/view/exercise-667080

Here's most of the part about weightlifting:

To get calorie credit for wieghtlifting go under cardio & search for "Strength training (weight lifting, weight training)". It's not much of a credit, but it's better than nothing (which is what you get if you enter the work under "strength training"), & it's easier than trying to find every exercise.

There are 3 goals or types of weight training.
Most people should start by building muscle.
After a month or so of that, you can add in building endurance.
Going for power is a more advanced technique, which I'll leave to the competitive weightlifters.

Free weights or machines?
Machines reduce the chance that you can do something wrong & injure yourself, don't take as much understanding of what you're doing, and target specific muscles or groups.
Free weights work more muscles in concert and allow for more exercises to be done in a smaller area, even at your home, but require learning how to do them correctly.

No matter which you're doing, you need to know the maximum amount you can lift, called the one-repetition maximum or 1RM.

For building muscle, you want to do 1 or 2 sets of 8-12 repetitions of a weight that's 70-85% of your 1RM.
For building endurance, do 1 or 2 sets of 15-20 repetitions of a weight that's 50-65% of your 1RM.

Either way, start low on both weight & reps and work up.
You should just be able to do the last 2-3 reps.
When it gets easy to do the maximum # of reps, add 5 pounds and go back to the minimum # of reps.

(From the American College of Sports Medicine's book "Resources for the Personal Trainer, 4th edition".)

Remember to work both sides of a joint (or the body) - if you're doing bicep curls, also do tricep extensions or dips. If you're doing quadricep extensions, also do hamstring curls. If you're doing abdominal curls, also do lower back extensions.