^ Tasha Yar's reference to the Enterprise-D as a "Galaxy-class battleship" in Yesterday's Enterprise seems to agree with that idea.
Regarding earlier large ships, the dreadnought USS Entente was mentioned communications chatter in TMP. The Kelvin's class is also fairly large - roughly 450 meters long, IIRC, as is the Excelsior-class - roughly 700 meters long (required by the hole in the Enterprise-B in Generations, and supported by the MSD, some design irregularities if the ship were of a smaller size, some side-by-side appearances with other ships, and the Enterprise-B's MSD). (The Nebula-class and Sovereign-class are also fairly large)
On screen, we've seen or heard ship classes described as follows:
Explorer/Battleship/Dreadnought:
(Entente)
Galaxy
Heavy Cruiser:
Constitution
Ambassador
Light Cruiser:
(Drake)
Destroyer:
Saladin
(Centaur)
Frigate:
Thomas Paine
Renegade
The Deep Space Nine Technical Manual also describes the Nebula class and Excelsior class as explorers.