I say take the coach : direct, quick (1h30) and cheap, about £18 return (but you will probably get it for £12 if you look out for special offers). I have never had trouble with luggage but I always limit myself to a single case that I can handle without help. See http://www.nationalexpress.com
When you get to Southampton catch a taxi from the bus station to the ship, costs about £6.00 each way.
The Cunard transfers cost £40 each way and I have heard varying reports (should be perfect for that price!). This is over twice the cost of above.
Going by train is more expensive and more bother.
The other idea is to use a private car / limo service if you have a large party to make it worth while. I don't have any specifics on this for you.