Did Tolkien mention anything about other planets in his mythology or did he invent just the one planet for his stories?
No other planets are visited in the Legendarium1, but several of Tolkien's dictionaries (published in History of Middle-earth in a number of places) include Quenya and Sindarin names for what we now know of as planets2; whether they were known to be planets by the inhabitants of Middle-earth is unknown.
- Morwen (Gnommish) and Silindo (Quenya): Jupiter (from The Book of Lost Tales, Part 1)
- Carnil (Mars)
- Luini (possibly Neptune)
- Nénar (possibly also Neptune)
- Lumbar (Saturn)
- Alcarinquë (Jupiter)
- Elemmírë (possibly Mercury)
The final examples are from the Index of Morgoth's Ring.
As TGnat reminds me in comments, the star known as Eärendil corresponds to what we would call the planet Venus. The complication here is that we know what Eärendil is: it's a half-elf (named Eärendil, funnily enough) flying through the air with a Silmaril strapped to his forehead, so it's definitely not a planet.
Since no further details about the other "planets" I've named above are given, it's entirely possible that they are also not actually planets in Tolkien's cosmology, but some other kind of thing. Unfortunately we don't know for sure.
1 Except Aman, which is sort of another planet after the Fall of Númenor
2 This pre-supposes that Middle-earth is in our solar system, which was Tolkien's intention. We have a number of questions about that on this site, including How exactly is Arda supposed to be an ancient Earth? and Is Tolkien's Middle-earth in our Universe? and In what time period does The Lord of the Rings take place in the real world?