If you want to lose FAT as your primary objective, personally I would do things a little different.
You have an hour let's look at the best possible way to utilize that hour to maximize your fat loss and muscle growth potential:
A warm up is fine, 10 minutes on the elliptical, bike or treadmill will get your blood pumping. I would spend no more than 10 minutes warming up.
So you understand that you can actually achieve better weight loss results by strength training as your primary routine without interjecting bits of cardio into the mix?
Strength training is primarily anaerobic, which means it doesn't require a lot of oxygen or endurance necessarily to complete a routine, you're going to increase the muscle mass in your body by working out at a higher intensity, which will in turn create a more suitable environment for fat loss. The more muscle in your body the easier it will be to drop fat.
I would incorporate a full body split for those three days (monday, wednesday, friday) (chest/back, legs, arms). 45 minutes in a solid routine, little rest, you will see results.
Math - Science
Get off the elliptical and get over to those free weights, they will prove to be a better use of time, and here's why.
If you run a 10 minute mile @ roughly 100 calories per mile, in one hour, you would burn 600 calories, right?...RIGHT!?!? Probably...but would you gain any muscle in the process? It's possible, although it would be primarily in your legs and core.
If you are conditioned to do that that's great, and is what some people prefer, however if you are looking for a better physique, possibly that of Mensfitness magazine or that of a body builder, cardio is not to be your primary weight loss or strength building strategy.
If you work out for 30-45 minutes you would burn between 300-500 calories with a med/high intensity. You would gain muscle, and actually work out muscles in your entire body, you would become stronger, as your strength increases running that mile would become easier, your body would condition itself, and then if running is still your goal, you can practice from there.
You say your goal is to lose weight and then gain muscle, but in reality, you would probably want to gain muscle first and then lose the fat. Once you have the muscle it will be much easier to drop the fat, oppose to dropping all kinds of weight, becoming a twig, and then trying to get muscle back (which in the process would put fat back on as well). Just do it all in one and strength train!
Don't forget your diet, that's actually more than half the battle, but that is not in your question so...