I have a big water temperature change when I have the shower running in down stairs bath when someone turns on the water in the upstairs. If they turn on cold, the temp goes hot. If they turn on hot, the temp goes cold.
I have 1/2" all the way through. Would a 3/4" header on hot and cold branched off to 1/2" help? Or could I go bigger than 3/4" and see an improvement?

Without going into actual pipe sizing code requirements, each bathroom should have 3/4" branches going to them, at which you are able to reduce down to 1/2" at the first fixture served, preferrably the tub/shower, 1/2" to the toilet and 1/2" to the lav sink. You may have a volume problem long before the bathroom. If your incoming cold service line is too small, this will equally affect both the hot water system and the cold water system. The hot water system should run 3/4" to the furthest bathroom with 1/2" branches for kitchen and laundry. The same with the cold water. If you have a two bathroom house, you typically need a 1" cold water service from the meter or well to at least the water heater. If this is not the case, you should upsize your water service first. This gives the most bang for the buck. If you have galvanized steel water piping, chances are you will need to replace most of it to restore proper volume because of rust and build-up. Because one fixture is robbing the other, it is a volume problem and not a pressure problem, although low pressure adds to the problem. Now, if either tub/shower/faucet is an older style single-handled Moen, it is possible that they are mixing improperly allowing cold water to flood back into the hot system and vice/versa. You can buy Moen replacement cartridges at any hardware store. Just specify old style or magnum. Do this first if it applies before you start replacing pipes. Hope this helps.

How old is the house?
What kind of plumbing do you now have? Iron pipe, copper?
Size of water main coming into the house?
Size of pipes leading to the bathrooms from the water source? 1/2" 3/4" other?

What Askaplumberfirst has a lot of good advice we may be able to pin down your problems better knowing this information.