If you go to http://www.britrail.com/index_us.html
you can get information about schedules. The ticket prices are for Britrail passes, which are very economical if you're traveling around Britain, but if you're just going from London to Southampton (and back) it probably won't be a good deal. You'll probably just have to show up at the train station and purchase a ticket.
There's a book called Britain by Britrail
that may be worth flipping through at a bookstore to find info about that particular journey.
I've ridden the trains in Britain many, many times, and there isn't anywhere to check your luggage. You have to carry it into the train car and then place it on or under a luggage rack (or on a rack above the seats.) I've done it with suitcases both light and heavy - just be sure you can maneuver your suitcase up and down three steps or so (to get on and off the train.)