It is entirely possible to get from the Houston Amtrak Station on a Houston transit bus.
But you would need to get an eastbound bus on Washington ave. (where the Amtrak station is), take it a few blocks, then transfer to a southbound bus to the greyhound station.
However, I would never do this. It wouldn't be real easy if you have a lot of luggage.
Also, the Amtrak station in Houston is in a not-so-good area of downtown,
and I wouldn't want to be walking around there even in daylight.
Taxis are abundant in Houston, so I would recommend just taking a taxi to the Greyhound station.
I have done that several times. The taxi ride is about $6 or $7.
Also, I just looked on Amtrak's site, and there is connecting service via motorcoach in both directions into Galveston at Longview.
However, it is very difficult to get around Galveston if you don't have a car there. There is some public transportation, but having a car would be preferrable.
From Houston, it's an easy drive to Galveston, less than an hour.
'Hope some of this helps.
David