I need to get a train from airport to Stratford?

Heathrow airport to stratford upon london & i'm thinking of taking a train. National Rail & CHiltern seem to be the only options. Are they big enough to fit large suitcases? Is there someone who loads it for you? Or do you just have to keep it by your side. You know, I think I'm imagining the Harry Potter trains to wizard school. Are there any large trains that go directly to Shakespeare's birthplace? I don't mind taxiing to somewhere not too far from the airport.