I know there is the "Seven Mounting Heavens of Celestia" and the "Nine Hells of Baator" How many outer planes are there, and do they all have long names like this?
Yes, they do all have long names. :) The seventeen outer planes are (listed on Manual of the Planes, p. 28) as follows. You can click each one for some information about it.
- Heroic Domain of Ysgard
- Ever-Changing Chaos of Limbo
- Windswept Depths of Pandemonium
- Infinite Layers of the Abyss
- Tarterian Depths of Carceri
- Gray Waste of Hades
- Black Eternity of Gehenna
- Nine Hells of Baator
- Infernal Battlefield of Acheron
- Clockwork Nirvana of Mechanus
- Peaceable kingdoms of Arcadia
- Seven Mounting Heavens of Celestia
- Twin Paradises of Bytopia
- Blessed Fields of Elysium
- Wilderness of the Beastlands
- Olympian Glades of Arborea
- Concordant Domain of the Outlands
The full table on Wikipedia has shorthand names for them and some base information as well.
Finally, the entire cosmology talks about how those fit in with the other planes. This beautiful map illustration of the Great Wheel may also be useful; it is based entirely on canon sources and quite accurate.