I am 47 years old & have had the above symptoms for as long as I can recall. I learned approx. 7 years ago that I have connective tissue disease (specifically Lupus). The symptom that you describe may be linked to connective tissue disease. During my first visit to a rheumatologist, an intern was working in the practice at the time. The doctor called him into the exam room to view my skin, which becomes particularly purple on my limbs, not so much on the trunk of my body. It is not extremely common so apparently it was a good opportunity for the lucky intern to view a case such as mine. There is a name for the condition but I don't know what it is. My skin as always appeared comparitavely "sheer" to me, since childhood. It's not like you can see all of my veins through my skin, but on my breasts and the inside of my arms, you can. The "purple effect" is absolutely temperature related for me; it disappears in the heat/warm bath/warm shower, however it does not have to be extremely cold to become prominent. 60 degrees will do it. It looks like a map of thick lines of purple, all interconnected, with pink or white skin in between the purple areas. Almost like a sloppy "honey comb pattern" is how I would describe it.
In any case, I would recommend that you consult with a good rheumatologist. Do not accept the opinion of a general practitioner as it is likely a symptom they will not recognize. Not a single doctor, my entire life, correctly knew what it was until I was 40. Then the connection was made between the skin condition and CTD. Watch for discoloration of your finger tips and toes in cold weather (blue, white or purple "ish") and pain in the above-mentioned areas when very cold, as this is also a related condition, Renauld's Syndrome.
PS: To the 13 year olds, I know how you feel. Purple ("Barney Legs" as I call them) can be embarrassing. Self tanning products help. If you are going to be in air conditioning, try skipping the shorts, wear jeans. My legs were always purple in gym. No getting around it. The most important thing is to get yourself checked-out by a good doctor... rheumatologist or other excellent diagnostician in your area. The first hint of my Lupus was noticed by an infectious disease doctor. They tend to be good at diagnosing complicated conditions so that is an option too.

Im 18 and get this too! i wish someone would just come up with a remedy just so in the summer i can wear a skirt and shorts with bare legs like all the other girls!
When i do use thick moisturiser it does help alot but doesnt last long.
i wish there were any tablets or special cream that would stop it forever!?

yes it is circulation problem the veins get contracted due to the cold and is causing the mottling of the skin. you need to avoid cold to prevent the damage to the skin

i'm 13 and i get that as well, it gets better in the warm but it can still get red and purple patches. How can i stop this?

I'm also 13 and i get this all the time, its not too bad but my legs go pale and i get purple and blue blotches on my skin, like my skin is see-through and i can see my veins. I do a lot of netball, running and gymnastics and I'm constantly in shorts and its really disgusting and i cant prevent it by staying warm because even when i'm not cold i get it. i Really need help with this, its so embarrassing. By the way, i find moisturising my skin and rubbing it helps it go away