Switching between ISAs should be straightforward. As some providers don’t allow transfers in, first check that you can move your money to the new provider and that there aren't any penalties for leaving your existing provider. Once you're happy that you won't get stung for transferring, contact the new provider. They will give you an ISA transfer form to complete, which will include details of your current provider. Once they have these details they'll contact the provider and request the transfer on your behalf. Whatever you do – don’t withdraw the cash, once outside of the ISA wrapper you will not be able to reinvest.
It should take less than a month for your money to turn up in your new cash ISA and many providers will pay you interest from the day your money leaves your old ISA provider so you're not out of pocket.
You don't need to transfer your entire cash ISA either, unless it's money that counts towards this year's tax allowance. This has to be transferred in its entirety.
With previous years' ISA money you are free to divide it between providers. So, for example, if you paid £3,000 into an ISA two years ago you could transfer £1,500 into one ISA and the remaining balance into another.