I had awful pain when I first started Copaxone so I know just how you feel. Here's a few tips that helped me.
Make sure your injection is body temp before doing it (hold it in your had or stick down your bra for a bit).
Try using the heat pad/cool pad thing you get. I found a warm one before and a cold one after worked best but I tried various, 2 warm, 2 cold, cold then warm, warm then cold. I could just use the cold pad afterwards quite quickly and still use it when I get the odd time that stings/hurts.
Try taking an antihistamine (just the 1 a day ones). Bad reactions can mean that you're slightly allergic to it, so can help until your body gets used to it.
I had been on Rebif before and manually injected (the noise of the auto thingy make me jump and pull the needle out too quick). I really struggled to manually inject with Copaxone because the pain was almost too much, so I tried the auto injector as well. Once again I found I jumped and pulled the needle out too quick, also it didn't help with the pain or site reactions.
SO.....I ended up getting some Emla Cream from my GP (a nurse friend recomended it). It's the stuff they use to make the skin numb in hospitals. You need to get the plaster things to stick over it on prescription as well. It's a bit more messing about as you put it on about an hour before you inject but it made such a diffrence to me.
I ended up allergic to the plasters that you put over the cream so stopped using it, but by that time my body was used to it and things had clamed down loads so I just needed to use the cool pack afterwards for a while.
Sorry if I rambled on a bit too much for you but I know exactly where you're coming from. I've now been on Copaxone for about 3 years and I'm glad I stuck with it as it's doing it's job well.