My wife and I moved to Wuhan from Canada a year ago and are enjoying the city and the culture.
If you are visiting from abroad and want to mingle with the locals, Mr. Mai's Coffee House and Foreign Language Club in Wuchang is a great place to visit. You can enjoy brick-oven fired pizza in Gianno's restaurant on the first floor and then walk upstairs to Mr. Mai's. As soon as you take a seat, you will be approached by numerous locals that will ask to join you to practice their English.
Another great site to visit is the Wuhan Botanical Gardens, a 70 hectare showcase of central China's plant life. There are also numerous special displays such as medicinal herbs and a huge bamboo garden.
Mo Mountain is another nice place to spend a few hours. There are many walking trails and free heritage music and dance shows every day at the display center at the top of the hill. You can walk up or be pulled up in a small cart. How you come back down is up to you; walk the trail or ride the cart down a mountain track to the bottom at breakneck speed!
Tailor Street or Yan Zhi Lu is another wonderful place to spend a couple of hours. The street is lined with clothing shops. You can buy off the rack but if you are around for long enough, you can get anything you want tailor made and come back and pick it up at a later date.
The local food is incredible but if you are longing for Western menus, check out Cafe Brussels or the Aloha Diner. The former offers great steaks and a large selection of international beers. The latter has great hamburgers and fries, real milkshakes, and Tex-Mex food along with a variety of standard American meals.
Wuhan is not a tourist city, a little rough around the edges, and probably has some of the craziest drivers in the world (its worth a taxi ride just for the experience) but you will enjoy it immensely.