Some of your routine may be dictated by how crowded the gym is, and how long the lines are at each machine. You often can't just hang around the machine waiting to do your next set (at least not if you're polite), but have to go to the back of the line and wait for your turn again. (But if someone is monopolizing a machine, you have the right to ask if you can 'work in'--do a set while they are recovering).
I would probably do a pyramid-up type workout, increasing weights (and probably decreasing reps, but doing reps to failure) on every set. In most cases, your first set may be a bit light anyway, as you adapt to the machine (if they're like the annoying ones that just label the stacks by a number rather than buy poundage). For poundage, you will be able to lift more with a machine than with freeweights, as the machine does some of the stabilizing that your muscles would do when working with free weights.
A possible routine (doing as a push-pull or as straight sets):
After a warm-up, do squats using the squat rack, lat pull-downs, bench press. (you could also do a set each of leg extensions and ham curls before going on to squats).
Repeat this once or twice, then go on to a second circuit doing leg press, seated rows and cable flyes or chest machine (forget what it's called). On the latter, watch for shoulder positioning, many chest machines have the pads set back too far, so the elbows start behind the body, which puts excessive stress on the shoulders. Adjust the machine so the elbows are a bit forward of straight out to the side, or even a bit farther forward.
Then go to shoulders and arms: shoulder press, bi-curls, triceps pressdown from the lat pull-down station. Second circuit: another shoulder exercise (depends on what them have--maybe use freeweights here), another bicep exercise, another tricep exercise (you might want to use freeweights for these as well: maybe preachers curls for the biceps, using a preacher's bench).
Finish with abs/core (maybe make use of a hyperextension bench if they have one) and a stretch.
