Albany International Airport: A taxi or rideshare is the sensible airport transfer: roughly 15–25 minutes to downtown, about $35–45 before tip. CDTA Route 610 does serve the airport, but it is slow and its evening/weekend frequency is poor; use it only if the timetable happens to fit, not as an assumed airport link.
Albany–Rensselaer Amtrak Station: The station is across the Hudson in Rensselaer, not walkable to downtown Albany. Take CDTA Route 214 to downtown when it is running (roughly 15–25 minutes; local-bus fare, about $1.50 cash or less with Navigator), or use a rideshare/taxi for roughly 10–15 minutes. Avoid casually hailing a cab: station taxi pricing has long been a poor deal, so confirm the fare before getting in.