Definitely Manchester. There's a frequent direct train service to Sheffield from there and it's smaller and closer than London. The train tickets from Manchester Airport are also cheaper than from London. Check out this link to the National Rail Enquiries to see timetables and fare prices.
Also, if you're planning to use the train a lot whilst you're in the UK and you happen to be a student with a valid Student Card, bring it with you and a passport sized photo so you can get a Student Railcard. This will cost a little at first but you can get 1/3 off the price of every ticket you buy. You can always get this card at any time provided you're a student or under 26 (I think!)