It is so important to get the right trainers if you are running or are going to properly.
I have Asics, and change them every year. Just about to buy my new pair today.
If you go to a Runner's Need or somewhere similar, you will get a gait analysis and they will point you in the right direction.
Personally, I would only ever run in Asics or New Balance - fashion trainers are not ones to run in.
I would also recommend double skinned socks as they will help you avoid blisters.
If you've not done any running before, then a good couch to 5k guide is a good start.