Hi Normandie, nice to see over here. Well first of all welcome to my city. From Vigo you could get a taxi, but itīs rather a long way, to say the truth about 88 km, so fares could be abusive.
My advise is relying on local trains. Fast, relatively confortable and hourly to Santiago. It takes you about 15 dollars return for a 90 minutes journey. For timetables check at
Train station is the northern part of Vigo, and you would have to walk up for 20 minutes. So get a taxi to the station. While in Santiago, old town is just 10 minutes walk from the Station.