If you find a new ISA you like then go to that company and they will arrange for it to be transferred from the old companies ISA to the new companies ISA. They are used to doing all this as people move ISAs around all the time.
That way, as has been said, you dont lose the ISA tax free status.
If you remove the money yourself you lose all tax free status. If you reinvest it you will only get tax free this year on about £5,000 of it.
Newspapers, like the Mail on Sunday, have financial sections where they give advice on best ISA deals.