Cash/Mini ISA the way to go then I'd say. Most banks do them, as do various online banks etc. Do a google search for best Mini/ Cash ISA rate.
If you have a mortgage and you'll be saving regularly you could also switch to an offset mortgage (like Intelligent Finance) and instead of earning interest it's taken off your mortgage, so you pay it off quicker without making an effort, you can also easily pay off lump sums (eg £1000 every 6 months or so)