Well, at 5'9" or 5'10" and 135lbs, it would be impossible to be "fat".
Do you want to go to the gym or work out at home? Either way, here are a few exercises to get you started. I would start with basic exercises like these. Once you are comfortable with your body's limits and proper exercise technique, you could add in more exercises. Go to the sporting goods store and buy pairs of 5, 10 and 15 lbs dumbbells. That should get you started.
Arms/Upper Body
Pushups: Classic exercise. Keep your back straight like a plank. Do them from your knees if necessary. Works you chest, triceps and shoulders.
Shoulder press: Sit in sturdy chair with back straight. Pick up whatever weight of dumbbell you are comfortable with and press straight overhead. Bring back down to shoulder height and repeat for 8-12 repititions. Works shoulders and triceps.
Bicep curls: Stand or sit in a chair. Pick up a comfortable weight and curl each arm. Do 8-12 repititions on each arm. Works biceps (obviously).
Legs/Thighs
Squats: Stand with feet shoulder width apart. Squat down, bending at knees until your thighs are parallel to the ground. Do repititions until thighs burn. If you need more resistance, hold a dumbbell in each hand. Works thighs, hip flexors and gluts.
Lunges: Start with feet slightly closer than shoulder width. Step forward with one foot (further than a normal step) and drop back knee towards the ground. Come back to starting position and repeat on other side. Don't let your front knee go past your toe and don't let your back knee hit the ground. If you haven't done these before, it may take a little practice, but they are an awesome lower body workout. Again, with these if you need more resistance you can hold a dumbbell in each hand. I recommend starting with no weight.
Walking/Hiking: Walk or hike up steep hills. It is a great leg and cardio workout if you do this over hilly terrain. Hiking incoperates more muscles than walking over smooth terrain, particularly in your lower leg. Walking up steps is also a great leg workout. If you have a football stadium or a several story building that you can get access to, those are great places to do a stair workout. But you can do it anywhere there is a flight of steps.
Doing a workout of the exercises above, you could make a nice little 30+ minute routine. You can stretch it out to however long you like by changing the distance you walk/hike.