The park is really in the middle of nowhere - it's far easier to get to Alton Towers if you're driving, but it's really not impossible to get there via public transport.
Take a look at Alton Towers' How To Find Us
page. You can get buses to the park from surrounding towns / cities Stoke on Trent, Uttoexter, Derby and Nottingham. Out of these, I'd suggest that Stoke is the easiest option in terms of travelling up from London by train.
You may find www.thetrainline.com
useful for checking train prices - should be pretty easy to see where the cheaper but non-flexible (and reserved seat) tickets are still available. Buy sooner rather than later as there's only a small quota of the cheap tickets before they make you buy a more expensive ticket. As long as you get an early train up in the morning from London, you'll definitely be able to get to the park and get back in the same day - it'll just be a long day with an early start. But it's well worth spending a couple of days there if you have the time - make sure you check out the water park! I recently did the journey from Euston - Stoke and it's only just over an hour and a half. The staff bus from Stoke only takes 45 mins (Bus 15, although there is a bus 14 that takes longer as it stops off at local villages). Check www.traveline.info
for bus times - they're not so frequent and you'll need to make sure your train gets you up in time to catch one. When it says 'Alton Towers Farley Gates' as the bus stop on that website, ignore that - the bus goes right to the front of the park.