Jim:
No train that I know of but I think there may be a bus or limo service from Hobby Airport. Trinity cemetery is at roughly 45th street and Broadway. It is also the cemetery where the memorials to the Union dead and Confederate dead are located and has lots of Civil War sites (e.g., graves of Generals Magruder and Wigfall). The Rosenberg Library is at 24th street and most of the historic Strand area (including the two history museums) is walking distance from there. Lots to do in Galveston including a lot of really neat historic homes and a couple of museums (air and railroad) that you might not at first associate with the city.