I am planning to fly to Frankfurt and I saw that there are two airports: FRA (Frankfurt Airport) and HHN (Frankfurt–Hahn Airport). Which one is more convenient and closer to cultural places including Mainz and the Rhine Cruise?
FRA is about 15 km from the center of Frankfurt. HHN is about 125 km away. FRA is also closer to Mainz than HHN is.
Frankfurt Flughafen is the right choice for what you have described.
From Frankfurt Hauptbahnhof it's a 10 minute walk along the river to the Römer Platz (the cultural center of gravity) where there are riverboats waiting to take you on an excursion up the Rhine river to scenic locales like Rüdesheim am Rhein (wine tasting in season).
The same "S-Bahn" leaving from the airport can also take you west to Mainz and Wiesbaden.