In addition to the posts above you did mention an overnight stay, which would be in a B&B - this is what we always have done, we live a 4.5 hour car journey away and its much nicer to not have that at the start of your day.
There is a B&B right outside Alton Towers gates called the Trough Ivy House, I've stayed there a couple of times, cosy but nice and its closer to the turnstiles than the car park is! There is another 5 minutes walk down the road called the Chained Oak which we have also used twice, again very nice. Its a short walk up the hill but its a bit of a busy road with no pavement/sidewalk. I always see people doing it and its probably fine just something to bear in mind.
At a quick glance here are some bus times from Uttoxeter station, if you scroll down from the 5th timetable onwards there seem to be buses every couple of hours. I can't tell you how accurate this information is, although it was only posted a year ago. There may be a better timetable somewhere but this should work as a guide
Bus times Uttoxeter station to AT