Tianshui City was built up along a river and is thus a long, fairly narrow town split into two distinct city areas - Tianshui and Beidao (though they recently renamed Beidao Maiji, everyone still calls it the former)
Beidao houses the train station, and the busses to Maijishan leave from this side of town. Tianshui houses the main long distance bus station, as well as most of the tourist attractions. More money seems to head towards the Tianshui side, overall it's more modern and clean, and even if you don't hit the attractions there's simply more to see and do.
It's easy to get from one side to the other. A number of local busses make the trip continuously (leaving from the train station), or you can grab a taxi for 20 kuai. The trip takes roughly 45 or 20 minutes, respectively.
Favorite thing: Oddly, as most Chinese towns sell them at streetside newsagents or in bookstores, the only English Language maps of the area are to be found in the gift shop at the China Post Hotel (just northwest of the main square).