The Thomas Cook Rail Map of Europe shows ferry services operating to Palma from both Barcelona and Val�ncia. Of these, Barcelona is easier to get to.
The German railways web site has all European rail timetables:
http://reiseauskunft.bahn.de/bin/query.exe/en
You can seek multi-country journeys but you should be aware that simply asking for a journey between Manchester and Barcelona usually comes up with a suggestion that allows insufficient check-in time for the Eurostar connection.
As an example of journey times, here's how you could do it next Thursday. (This is based on times from that German website, modified to give you time for the Eurostar connection):
Manchester Piccadilly 1115
London Euston 1328
Euston to Waterloo takes about 15 minutes on the Northern Line underground service.
London Waterloo 1511
Paris Nord 1853
Gare du Nord to Gard d'Austerlitz takes about 25 minutes on Metro Line 5
Paris Austerlitz 2032 (Sleeper service)
Barcelona Franca 0824
So that's got you to Barcelona fairly easily, and without missing any sleep, in under 24 hours. You've then just got to get a taxi to the ferry terminal for the trip across to Palma. Personally, I wouldn't bother. Barcelona has much more to offer than Majorca: Excellent beaches, superb restaurants, lively bars, good quality hotels and enough attractions to keep you busy all summer.
Chris