Throughout editions, there has been the astral plane as well as other planes. Which planes are in the Dragonlance setting?
Yes, there are several other planes of existence in Dragonlance - which ones depend on your edition
The 3e Dragonlance Campaign Setting, on pages 117-118, gives a brief overview of the planar cosmology of Krynn. According to this resouce, some of the planes of existence that are standard to D&D cosmology can be found in Dragonlance, but the roster is rounded out by several unique ones too. In this model, the planes of the setting include:
- The material plane (unsurprisingly)
- The 4 "inner" elemental planes:
- The elemental plane of Fire
- The elemental plane of Water
- The elemental plane of Air
- The elemental plane of Earth
- Three "outer" planes:
- The Dome of Creation, home of the gods of Light
- The Abyss, home of the gods of Darkness
- The Hidden Vale, home of the gods of Balance
- The Astral Plane
- The Ethereal Plane
- The Shadow Plane
- Unspecified extra planes that are the realms of the gods of magic
- Various demiplanes found embedded in places throughout the previously mentioned planes
The more "standard" outer planes described in generic D&D resources, such as the Nine Hells, do not appear to exist in this version of the Dragonlance setting. Unlike the default cosmology in which the outer planes are more distinctly separated from the inner planes - and so, for instance, the ethereal plane does not extend to the outer planes - the cosmology described here seems more tightly bound together, with the Ethereal, Astral and Shadow planes reaching both the "inner" and "outer" planes. (In fact, the nomenclature of these groups gets a bit confusing as the "inner" planes are described as encircling the "outer" ones!) All of the above is set within the unimaginably vast Ethereal Sea, of which the world and planes of Krynn are just one small island amongst countless others.
As ZwiQ's answer notes, the cosmological model presented in 3e is a significant departure from previous editions, where the standard Great Wheel cosmology seems to have been assumed by default and the gods were assigned homes on the usual planes of existence. Dragonlance doesn't have any dedicated setting books published since the 3/3.5e era, so no subsequent sources have directly overruled 3e's description of the planar cosmology of Krynn - though based on the developments of other D&D campaign settings in the same period, it's quite plausible these changes would be ignored or retconned if newer material was published.
The Standard D&D Cosmology Applies
The standard D&D cosmology applies for Krynn as well, at least in a multitude of 2e, as well as 5e products. All Spelljammer and Planescape materials describe Krynn as they do any other world: located in a crystal sphere (Krynnspace) in the prime material plane. For example, On Hallowed Ground places Paladine's realm on Mount Celestia, Gilean's realm on the Outlands, and the realm of Takhisis on the Nine Hells.
The first chapter of the 5e Dungeon Master's Guide is about world building. In it, we read (page 9):
This book, the Player's Handbook, and the Monster Manual present the default assumptions for how the worlds of D&D work. Among the established settings of D&D, the Forgotten Realms, Greyhawk, Dragonlance, and Mystara don't stray very far from those assumptions.
Later, we see Krynn listed amongst the Known Worlds of the Material Plane, at the end of Chapter 2, a chapter that extensively describes the standard Great Wheel cosmology.
But what about the description in the answer by Carcer? During the 3e, there was an attempt by the Wizards of the Coast to "simplify" the game by making the cosmology of each popular setting separate from the others. For example, the 3e Forgotten Realms Campaign Guide introduced the World Tree Cosmology specific for the FR setting. This attitude has been dropped in the 5e. Yet in-game, you could imagine that these alternative cosmologies represent the (broken/incomplete?) "views" of the sages of the individual worlds. It is also worth remembering that the Outer Planes are all about belief, so for those "clueless prime" sages, the planes can literally appear as they are imagining.