Is the East Square Railway station in Xian the same as the HST station; if not, how far away to walk to catch BUs 306 or bus 5. How frequent is the bus service.

No, they are different. HST Railway Station you mentioned refers to the North Station (Beikezhan), but the former refers to local main railway station with normal trains. It is 13-14km between. You can take bus no. 266 to travel between directly and the interval of it is 15-20min. Or you can take a taxi directly.

HST Station has in the same building as Beikezhan MTR Station of line 2.

To take a taxi to travel from Beikezhan to Xianyang Airport, it costs around CNY60.

Do you mean Shaanxi History Museum? If so, by bus, you can get off at Cuihua Road Station. Upon your getting off, you will easily find the museum.

Great Big Waild Goose Pagoda (Tayanta), you need to get off at Dayanta Station or South Square of Dayanta Station.