Don't eat too much (but also not too less. Eat well!), and run/jog for at least 40 minutes a day (make sure to run on grass or your knees will be stuffed, and run in the afternoon or evening and not in the morning. If you can find a route with some hills or some other incline that would be awesome) or do swimming.
To strengthen your core stay in the 'up' push up position for at least 3 minutes. I would refrain from sit ups/crunches.
To strengthen your arms and shoulders, do push ups, triceps dips, and assisted pull ups/ chin ups (these are called supine rows. When you can do at least 50 in good form progress to actual pull ups/ chin ups on a rafter at home in good form. Make sure to keep your grip no farther than shoulder width apart or you're going to put too much strain on your rotator cuffs and dislocate/sublux one of your shoulders--not cool.). Make sure to go on youtube and watch videos teaching you how to do these exercises in good and proper form, which will minimise your chance of injury.
You can do all the above in circuits. You can design your own circuit based on your current abilities from the following site-
http://www.royalnavy.mod.uk/custom/navy/trainingtool.html
Try to stay as motivated as possible in any way you can (you're not tired until you hit the wall, and your limit is death. Pain is temporary and is weakness leaving the body, and so on), and be flexible with your fitness plan (plan to complete certain exercises by the week. Don't limit yourself to doing a particular exercise on a certain day and not on any other as plans always go to shit.) or else you will lose motivation and forget it all.
You don't need to join a gym to be fit. Paying for gym membership can be quite expensive too.
All the best with it.
