While losing weight as whole will reduce the fat on your legs and arms, I am assuming you are more interested in keeping the skin on your arms 'tight'? What I am doing to target that upper arm area is doing weight lifts. I started off at 5lbs, and now can do 10lbs and it's been less then a month-So you can really get progress going if you work at it.
I am not really sure what the appropriate name for the lifts I do are, but I can describe to you a few different ones I've been doing:
Standing with one weight in each hand, simply lift your arm up towards your shoulder, and back down again. Doing it one arm at a time, Right, Left, Right, Left etc..Only count '1' when you have lifted both arms once IE: Lift with the right arm, lift with the left arm=1 rep not right arm=1, left arm=2. I can do anywhere from 200-400 of these at a time.
Sitting with one weight in hand, do the same lift style only going down between the legs so your arm is reaching straight outwards. Don't rest your arm against your leg. I do 50 with one arm, then switch so I am giving my arm a break and continue with 50 on the other then switch back to the other arm. I do around 100-200 of these.
Standing with your arms to your side and a weight in each hand, bring your arms upwards (not forward) and lower back down. If you look like an insane person trying to fly then your doing it right >>' Only count 1 when you've lifted up and come back down.
Standing with your arms in front of you and a weight in each arm, open your arms as wide as you can and bring them forward again. Like a weird looking invisible hug.
Standing, or kneeling if you really want to (or you have a low ceiling..), with a weight in each hand bring one arm up then back down and switch to the other arm. Your raising your arms as if you want to answer a question, keep your elbows straight and do not count 1 rep until both arms have come up. Right, Left=1. Not; Right=1, Left=2.
For working on my thighs I have a bit of an issue in that I've discovered I have a problem with my ankles and until I can get checked out I am very hesitant to do jogs or running. If you do this however, run in place in two ways; with your legs forward so your knee's are in front of you (as if you can put your hands out and when your knee comes up it would touch your hand) then change to a 'natural' running position with your legs behind you. I haven't been checked out yet to find out why these moves do an incredible amount of debilitating pain on my ankles :/ but they really do a good job.
I also do squats, where you stand with your legs apart how you'd naturally stand and squat down as low as you can go. Hold your arms out in front of yourself to help balance, and if you can hold weights as you do this. I can currently do 100 at a time, but need to get to doing more.
A move I learned in yoga class was to come down in a 'push up' style position only keeping one foot down as you bring the other leg upwards in a bending motion as if you where running facing the ground and continue switching legs in this running motion. Bringing one leg up as far as you can, back down and bringing the other back up while maintaining a 'push up' style position with hands flat and toes on the ground.
You could also try what I believe are called 'lunges', in which you are essentially doing the squats I described only one leg at a time as you hold weights.
I started off just a couple weeks ago barely being able to do 25 of any of these, now I am able to do several hundred in a day if I push myself hard enough.