http://www.moneysavingexpert.com/savings/cash- isa-transfers
This will help.
Quite simple if the new provider accepts transfers , but it can take 3 weeks and your money may be between providers for a week so you lose a week's interest. The banks sort out the transfer.
The important thing is not to withdraw the money yourself and close the account as this will cause your savings to lose their tax free status. Contact either your existing provider or your new chosen one to ask for an ISA transfer form, complete it and give it to your provider. Some provides have a facility allowing you to downloan and print off an ISA transfer form from their website.