Agree with gloves. I used to think football players were wimps, when they wore short sleeves and gloves, but they really do work. If you get hot easy but have cold hands, then that is the answer, that's for sure.
The Karrimor range at Sports Direct is cheap and cheerful. I like the Adidas range, a bit pricier, but it is better quality.. Always always wear breathable.... wear anything else and you will get too hot once running.
You want to be starting your run cool, (I think that has been said already) as when you get going, you quickly warm up. Don't wear too much as you will boil under all the clothes. Wear layers til you find what's best.
As you get better at judging the weather, you can adapt what you wear. I never wear a jacket, and wear as little as I can get away with in the winter, layers usually.
And do remember that skin is waterproof! You may find that a cool run in the rain to be exhilarating!
One thing. You started training in the colder months, and we have had unseasonal warmer weather on and off in the past month.
I find that my body takes time to adapt to the warmer temps. So when it is April and warm, its v hard to run in warm weather. But, as the months go by, the body slowly adapts to the warmer weather, and it does get better. Come July, yes I do find it hard, but it is easier than when it was April.
Just stay with is, allow yourself to go a little slower when its warmer, and your body should get used to running in warm. Good luck!