In creative projects there's a dead thread called operation cardio which may be helpful.
I would recommend swimming to increase stamina and cardio. If that's not available, I'd use an elliptical before a treadmill. Treadmills can be bad for your knees, and the elliptical claims to burn 990 calories an hour when I'm on it, but it seriously can't be right.
Free weights are better for you than machines. You get the full range of your muscles and your movement is natural. It is also slightly tougher since you don't have machinery to guide your movement. Pullups help your biceps and back, dips use your shoulder blades and triceps (the part from your elbow to your shoulder on the outside), bench press will help your chest and secondarily your triceps, free weight curls will focus your biceps, and the lat pull-down machine will help your shoulder blades.
Pushups are similar to bench presses, but you get a little more control of what area you want to work out. The closer you put your hands together, the further down your arm you work on. If your hands are close enough to have your thumbs and index fingers touching, you're working on your forearms and some triceps. A little further out would be biceps and triceps, normal spacing works your chest and biceps, and very far apart works almost exclusively on your chest and a little on your shoulder blades. Pushups will help you with your cardio.
I've been getting core results since I've started doing leg lifts correctly and crunches. Leg lifts work the lower part of our core, from your nethers to about halfway up your abs. Lay down on your back, place your fists behind your head so your shoulders never leave the ground, keep your legs straight and lift them off the ground using your abs. Resist using your back or thighs or it completely ruins the purpose, and will hurt your back.
To crunch, lay on your back, put your legs up with your knees bent 90* and crunch your abs up tight and hard. Instead of doing it like you see most people, so slow, tight, and really make your abs known to you. This prevents you from using your back, and you certainly know you're using your abs. You can also use an exercise ball for crunches, but personally they hurt my back.
Hope this helps, if you need any questions just holler. Fitness centers are not competitions, and no-one will berate you for starting out.