I have minimal qualifications to advise you, but I lost almost 70 pounds over the course of a year through exercise and sensible diet, I’ve maintained my weight for another year, and I now teach a step aerobics class. I’m not certified as a general fitness instructor or personal trainer, however.
Given your schedule, I’d advise you to start slowly and build up to working out as many days a week as possible. As most trainers will tell you, the best exercise is the one you’ll actually do, so I can’t really tell you which machines would fit you best. If you’re in really poor physical condition, you might start out with the easiest, which would be the treadmill, and then try out the others as you build strength. The Stairmaster and elliptical machines are likely to be the ones that will give you the most strenuous workout, but they’re also very difficult for the beginner. I’d suggest you start out with a goal of working out for 40 - 60 minutes a day, three days a week, on whichever machine you like best (or, when the weather gets better, perhaps outdoors). Warm up slowly for about ten minutes and work up into a pace that is challenging but not exhausting. Continue at that pace for 20 - 30 minutes, then step it back and cool down for another ten minutes or so. When you’re in the middle of your workout, you should feel like you’re working pretty hard. The gym should have heartrate charts you can use if you’d like to take your pulse to check the intensity of your workout, especially at first, but in general you should feel like your level of exertion is about 7 or 8 on a scale 10 (10 being as hard as you could possibly work). You should be able to speak, but not at length - if you can say short phrases without gasping but you wouldn’t be able to have a detailed conversation or sing, that’s about right.
You say you’re not really interested in strength training or muscle tone, but building muscle goes a long way towards helping you to lose weight. If there’s any way you can get yourself to do even a short weight routine on the days in between your cardio workouts, you may be surprised at how much good it does. You can do a minimal yet effective routine in just 15 or 20 minutes. The most efficient use of that time, since you’re interested in weight loss rather than, say, arm and shoulder definition, would be to do exercises that build the muscles in the legs and buttocks. Strengthening your quadriceps muscle (the front of your thigh) has the added benefit of stabilizing the knee joint and helping to prevent knee injury when you’re doing things like stair-climbing and running. Don’t do strength training on the same groups of muscles two days running. Give your muscles need time to rest and rebuild in between weight sessions.
I know it’s difficult to develop a healthy diet when you’re eating in cafeterias. There are steps you can take, though. Choose the freshest, least processed foods you can get. If you can, avoid anything that’s breaded and fried, bread products made with mainly white flour, and high-sugar items. You might make a habit of reading nutrition labels (some cafeterias even offer the information if you ask for it) and try to limit your calories. Whatever you do, don’t go under about 1200 calories a day. Your body will need calories to build muscle, and if you try to force it to run on stored fuel alone, you may just end up training your body to be more efficient in storing fat.
This site has information about the glycemic index of foods and what it means to your diet. And this USDA page has a lot of nutrition and weight-loss information, along with a whole lot of links to other resources. And I’ll go ahead and be the first to post a link to Stumptuous.com, which offers a wealth of information about working out.