If you are a little nervous about using equipment in the gym, perhaps you could start out with resistance bands. You can use them at home, at the gym, travel with them, etc. Sparkpeople has some comprehensive routines available on their website and you can check out YouTube for more.
I personally don't use any machines except the lat pull-down. I think free weights are a much more ergonomic way of strengthening muscles. However, I think with free weights it's important to get 'trained' by an expert who can show you proper form. (By trained I just mean go the first time with someone who knows what they are doing- the person running the gym, a trainer, a friend on an athletic team who lifts with a coach, etc) who can give you a demonstration. There is not a ton of room for error with machines, but there definitely is at least a bit with free weights. This is what makes them a little more intimidating at first, but in the opinion of many "better" for effective lifting.
As far as a routine, do some googling on Stumptuous.com, buy a Fitness or Shape magazine, etc. (the latter two are not amazing resources, but make strength training accessible to beginners). These will give you ideas for routines. You don't really need a bona-fide official "routine" you find somewhere, you can make your own based on exercises you see in resources like these. The most important thing is to hit all your muscle groups