First - you can still keep your domain registered at Heart ( I don't see any reason why you need to transfer the domain to another registrar) but turn off the free hosting & free email hosting, and change the domain DNS so email hosting is pointing elsewhere.
I have used Google Applications for a long time, and for free you get 7GB email accounts for your domain. You also get Google services free as well - Calendar, Sites and Documents, all using your own domain (don't confuse this with regular Gmail - this is something different). click here
Hotmail also do something similair called Windows Live Admin - again they provide email, calendar, office suites for your domain, but is not as good as Google, its fairly limited in features.
Beyond that, paid email hosting is your best bet - Fastmail, Tuffmail, EuMX, Rackspace to name a few charge £10 upwards to host your domain's email.
Heart Internet have one-click settings to enable your domain to work with Google Applications if you decide to go with Google Apps and keep your domain at Heart.
Hope this helps.