As far as I remember, when Aragorn and Gandalf discuss the alternative of Moria Aragorn says something like "I too have journeyed through Moria...". This would imply that Gandalf has walked the halls of Khazad-dum after the fall. So my question is, why, when and where can I read about it?
It must have been sometime between 2841 and 2850 of the Third Age (the events of Lord of the Rings take place in 3018). 2841 is when Thráin first started wandering, and 2850 is when Gandalf ultimately finds him:
2841 Thráin II sets out to revisit Erebor, but is pursued by the servants of Sauron.
2845 Thráin the Dwarf is imprisoned in Dol Guldur; the last of the Seven Rings is taken from him.
2850 Gandalf again enters Dol Guldur, and discovers its master is indeed Sauron, who is gathering all Rings and seeking news of the One, and of Isildur's Heir. He finds Thráin and receives the key of Erebor. Thráin dies in Dol Guldur.
Return of the King Appendix B "The Tale of Years" (ii) The Third Age
Unfortunately we can't be more specific than this.
To find Thráin, Thorin Oakenshield's father (emphasis mine):
'Good, Gimli!' said Gandalf. 'You encourage me. We will seek the hidden doors together. And we will come through. In the ruins of the Dwarves, a dwarf's head will be less easy to bewilder than Elves or Men or Hobbits. Yet it will not be the first time that I have been to Moria. I sought there long for Thráin son of Thrór after he was lost. I passed through, and I came out again alive!'
Fellowship of the Ring Book II Chapter 4: "A Journey in the Dark"
Where can you read more
Nowhere; sorry. Everything we know about Gandalf's journeying through Moria is revealed (in snippets) by his dialogue in Fellowship.
Gandalf's search for Thráin is elaborated upon very slightly in Appendix A, and in an essay in Unfinished Tales, but no mention is made of travelling through Moria to find him, let alone details of his journey.
I've searched through History of Middle-earth, and found no more detail than is presented in the published Fellowship.
Confirmation that it was one time only:
I fear we shall not reach them at all. It is not far, but our path may be winding, for here Aragorn cannot guide us; he has a journey in the dark 391 seldom walked in this country, and only once have I been under the west wall of Moria, and that was long ago.
Fellowship of the Ring pg. 390-391
The Mines of Moria were vast and intricate beyond the imagination of Gimli, Glo´ in’s son, dwarf of the mountain-race though he was. To Gandalf the far-off memories of a journey long before were now of little help...
Fellowship of the Ring pg. 405
I find it very hard to believe that Gandalf went though Moria only once before LOTR. I believe that Gandalf arrived in Middle-earth and started making his travels almost a thousand years before the Dwarves were driven out of Moria by Durin's Bane. For much of that time things would have been peaceful enough around Moria for the Dwarves to feel safe enough to charge money for travel though Moria, bypassing the dangerous Redhorn Pass.
Thus Gandalf should have passed through Moria many times on his travels, even if the last time was more than a thousand years before LOTR.