Oooh my specialty! (I'm an Excercise Physiology major so we spend all day studying the effects of excercise on the body.)
I haven't been to the gym much recently, but that is because of winter break and joining the hockey team (I'm sore enough from hockey 3 times a week that the gym would be overkill (although I do need to get back for some strength training)).
Anyway, you didn't really say what you do at the gym...
When I go, I stretch for maybe 10 minutes, then jump on an eliptical machine for 10-15 minutes. Then I usually do the nautilus machines until I'm either my muscles are finished or I'm out of time. When I'm doing what I should be doing, I'll have saved 10 minutes for a cool down stretch, but usually I don't.
I don't know exactly what you do, if it is a class or swimming, then I wouldn't worry too much about it because it is hard to over exert yourself swimming and in a class there is a trained professional there to tell you if you are doing something where you might injure yourself.
If you are doing something on your own (like what I do above), it is important that you know the right way to do excercises because you could get injured otherwise. If you can, your gym might offer something where someone can walk you through a routine once or twice and show you the right way to stretch and use certain machines. Or if you can't do that (it probably costs at least a little money to do that) you can look up most of the excercises online.
Anyway, forgive me if I got a little preachy, it is sort of something that I'm passionate about because excercise can be so much easier than people make it out to be. With a little practice, it just becomes part of your life and you won't even really notice when you pick up your gym bag on Tuesdays.
I'm finally back for the summer, although hopefully I'll spend a little more time away from this screen.