the lord of the rings - When does Gandalf learn the nature of Durin’s Bane?

Durin's Bane, the Balrog of Moria, was unearthed at some time in the Third Age.

Shortly afterwards, the city was abandoned, and no one with any sense came that way again for a long time.

As stated in this answer, it is clear in the LOTR films that Gandalf knew the Balrog was there. I am not sure if it is so clear in the books what he knows of its nature.

If he does know of the Balrog in the books, is there any indication when and how he finds out?

The nature of Durin's Bane seems to have been a bit of a mystery for most of the Third Age; the elves do not know. Obviously, the dwarves think that it has gone; otherwise they would not try to retake Moria.

I've checked the LOTR wiki and skimmed The Silmarillion but have not seen anything obvious about this.

Answer

Not until it's already upon them

Emphasis mine:

'What happened away up there at the door?' [Gimli] asked. 'Did you meet the beater of the drums?'

I do not know,' answered Gandalf. 'But I found myself suddenly faced by something that I have not met before. I could think of nothing to do but to try and put a shutting-spell on the door. I know many; but to do things of that kind rightly requires time, and even then the door can be broken by strength.

'As I stood there I could hear orc-voices on the other side: at any moment I thought they would burst it open. I could not hear what was said; they seemed to be talking in their own hideous language. All I caught was ghâsh; that is "fire". Then something came into the chamber - I felt it through the door, and the orcs themselves were afraid and fell silent. It laid hold of the iron ring, and then it perceived me and my spell.

What it was I cannot guess, but I have never felt such a challenge.

[...]

'Ai! ai!' wailed Legolas. 'A Balrog! A Balrog is come!'

Gimli stared with wide eyes. `Durin's Bane!' he cried, and letting his axe fall he covered his face.

'A Balrog,' muttered Gandalf. 'Now I understand.' He faltered and leaned heavily on his staff. 'What an evil fortune! And I am already weary.'

^{Fellowship of the Ring Book II Chapter 5: "The Bridge of Khazad-dûm"}

For what it's worth, in the text Gandalf himself proposes Moria as an alternative to Caradhras:

'There is a way that we may attempt,' said Gandalf. 'I thought from the beginning, when first I considered this journey, that we should try it. But it is not a pleasant way, and I have not spoken of it to the Company before. Aragorn was against it, until the pass over the mountains had at least been tried.'

[...]

'The road that I speak of leads to the Mines of Moria,' said Gandalf. Only Gimli lifted up his head; a smouldering fire was in his eyes. On all the others a dread fell at the mention of that name. Even to the hobbits it was a legend of vague fear.

^{Fellowship of the Ring Book II Chapter 4: "A Journey in the Dark"}