Women, weight training?

Weight training is an excellent idea but not after a run as your energy levels will be quite low, and (as far as I know), the reverse is actually better for fat burn (weights THEN cardio).

Maybe get a book like starting strength or New rules of lifting for women. You could even start with bodyweight exercises at home .

Check out the group: http://www.myfitnesspal.com/forums/show/788-women-strength-training

Make sure you do atleast 30-45 minutes cardio 4-5 times a week, and 4 times a week lift some weight. This can be on the same day righ afer cardio or you do the cardio in the morning and in the evining/afternoon you do the weight lifting.
Split up your routines to two: biceps, legs, back, calves and other day triceps, shoulders, abs, chest. 2 routines/body part, 3 series between 8-12 reps, with the weight you can manage.
You can look up plenty of dumbell excercises on the internet, youtube for example.

Weight training is good to maintain your lean weight, prevents you loosing muscle instead of fat, and muscles burn more calorie even at rest, than fat.

Sorry, but thats godawful advice, there is no benefit in split routines over compound movements which build functional strength. You also dont want to overtrain, as you wont see benefits from doing so. Better movements are things like dumbell squats, swings, deadlifts, Over head presses, bench presses, dumbell rows and push ups and pull ups. Choose one hip (squats or lunges) one push (eg bench press) and one pull (eg, rows) and depending on the weight d0 4-6 reps if heavy or 8-10 if light. Do 3-5 sets of each, 3 times a week

I agree with this advice. Isolation weight exercises, ie on machines or focusing on one muscle group only are not as affective as doing resistance work that engages a variety of muscles at one time- ie squats, lunges with weights, pull ups, dead-lifts, etc. Moreover, dont overtrain. With weight lifting it is PARAMOUNT that you have a recovery day to let you muscles repair and rest. Moreover, when you overtrain you seriously risk compromising form and increasing your risk of injury.
Also in response to something the OP stated. There are a variety of debates but I prefer running after weight lifting. Partly because the weights act as a nice warm up for the run, I can lift heavier and feel stronger when I start with weights first and find that if I lift heavy weights after a run my muscles are fatigued and I compromise on form- which is never a good idea. Also remember to increase your intake of protein!!!
As to how I train, I do a three day on- one day off work out and I run every day for 5km after my work out.

Good luck!! Find what works best for you, and dont expect to see results overnight. Building muscle is hard work!

I'm looking to tone up my whole body, especially my jelly belly and my arms, I have zero upper body strength! Is weight training a good idea?

Yes, absolutely.

I already run 3/4 times a week, would it be best to work in weight straight after a run (think I heard something about this burning more fat?)

I would either do weight training before the run (you will have more energy for the lifts!) or on separate days.

What kind of exercises do you recommend? I need something I can do at home with dumbbells as I don't have a gym membership.

While I would recommend a gym membership (they'll have more range than you could get at home, unless you had loads of money for equipment) However, if you can't for whatever reason there are ways you can improvise. Focus on compound exercises, they work your whole body. 3 full body workouts a week should be ok. Things like squats, deadlifts, shoulder press, chest press, rows etc. All of these, while usually done with barbells, can also be done with dumbells if that's all you have.
3-5 sets of 6-8 reps I always find it good for maximum results. The last couple of reps should be a struggle but still good form.

A good idea is an UNDERSTATEMENT!!!!! Every woman should lift....keep in mind it takes TIME.....I've been at it 20months and still a long way to go...diet is an essential part of it, you can build muscle till the cows come home but if you don't eat right you are wasting your time...good luck and hope you love it!