How do I get from Century City metro station to Shiyangchang bus station in Chengdu?

From Century City, you can take the metro to reach Fuhuayuan Station and then take a taxi to reach Shiyangchang Bus Station directly. The taxi fare is around 15 to 20RMB.

What about Wuguiqiao bus station? I am trying to avoid taxis

You can surely take a bus at Wuguiqiao Bus Station too, but it is much further from Century City than Shiyangchang Bus Station. If you wish to avoid taxi, I suggest you take MTR to Incubation Park Station (Fuhuayuan Station) first. Get out from Exit C and walk along Jincheng Avenue to Ruanjianyuan Station with about 500m to take bus no. 124 to Shiyangchang Station. After that, walk to the bus station to take a bus to Xichang.