The number 85 bus from across the road (outside Crewe Arms Hotel) of the train station and that brings you right onto campus. It costs roughly £2.30 but it may have recently gone up. It goes through alot of the countryside so don't panic if you think you might be on the wrong bus lol, its usually a mini-bus anyway, and is run by Arriva. This takes roughly 45mins as its a bus and takes forever.
From Stoke train station you can get the number 29 bus from across the road, and that will also bring you onto campus. It takes roughly 45mins (to drive it'll take 20mins lol) and stops in Newcastle-under-Lyme too, so don't get off there cos you'd be stuck 10mins away lol. Stay on until it enters the campus, and you'll know about it as its up and down a big hill and the uni looks like a little town as you come over the rise lol. There is a big medical school and several new science buildings are being built on the road up to the campus.
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And a taxi from crewe would cost around £12 as an estimate. Its further away and there isn't really any 'quick' way to get there. If you get the bus, you can get an earlier one and have a look around campus for abit?...