Also, I think that Gandalf knew about the Balrog, even in the book: spending that much time learning that much lore makes it a lot easier to figure out what Durin's Bane was. I think that Tolkein wrote him quietly wanting to have a showdown with the fiery fiend - if it had been available for the War of the Ring, things would have gone much worse for the free peoples.

You think Gandalf wanted to fight the Balrog?

If Gandalf wanted to fight the Balrog, why did he first try to go over the mountains?

Because he still wasn't sure of his eventual fate.

I don't mean he actively sought the Balrog, otherwise he would have gone straight to Moria alone after being rescued from Orthanc, but once the company entered Moria I think he realised where it was all leading and accepted that this was the only way to become the White Wizard that Saruman should have been. Bare in mind Saruman had already succumbed at this point so the Valar had no powerful Istari in charge in Endor, they needed Gandalf to be reborn. I've changed my mind that he knew the Balrog was there before he saw it, but he did want and need to fight it.

Remember Aragorn says, '...it is not for the Ring, nor for any of us others that i am thinking now, but of you, Gandalf. And I say to you: if you pass the doors of Moria, beware!' Aragorn felt that Gandalf had a destiny that had not yet been fulfilled.

Also when the Balrog appears Gandalf says to himself, 'A Balrog,' muttered Gandalf. 'Now I understand.' He faltered and leaned heavily on his staff. 'What an evil fortune! And I am already weary.' At that moment he recognised what fate had in store for him and why he had come into Moria.

It a bit like the story of Jesus and the resurrection, he knew he had to die to come back stronger, he didn't seek death but he accepted it had to happen. Same with Gandalf I think, Gandalf realised fighting the Balrog would destroy him but make him stronger to fight Saruman and Sauron.

Having said all that I disagree with TheOneRider that Gandalf was simply killing the Balrog to make the war of the ring easier. I don't think the Balrog would have been used by Sauron against Gondor.

'It [Moria] is a name of ill omen,' said Boromir. 'Nor do I see the need to go there. If we cannot cross the mountains, let us journey southwards, until we come to the Gap of Rohan, where men are friendly to my people, taking the road I followed on my way hither. Or we might pass by and cross the Isen into Langstrand and Lebennin, and so come to Gondor from the regions nigh to the sea.'
'Things have changed since you came north, Boromir,' answered Gandalf. 'Did you not hear what I told you of Saruman? With him I may have business of my own ere all is over. But the Ring must not come near Isengard, if that can by any means be prevented. The Gap of Rohan is closed to us while we go with the Bearer.'
'As for the longer road: we cannot afford the time. We might spend a year in such a journey, and we should pass through many lands that are empty and harbourless. Yet they would not be safe. The watchful eyes of both Saruman and of the Enemy are on them. When you came north, Boromir, you were in the Enemy's eyes only one stray wanderer from the South and a matter of small concern to him: his mind was busy with the pursuit of the Ring. But you return now as a member of the Ring's Company, and you are in peril as long as you remain with us. The danger will increase with every league that we go south under the naked sky.'
'Since our open attempt on the mountain-pass our plight has become more desperate, I fear. I see now little hope, if we do not soon vanish from sight for a while, and cover our trail. Therefore I advise that we should go neither over the mountains, nor round them, but under them. That is a road at any rate that the Enemy will least expect us to take.'

Just thinking about it, if it was too dangerous to go through the gap of Rohan, Moria, Caradras or the Northern wastes, why not head south and travel through perfectly safe lands? It would take a bit longer but it would certainly have been easier.

why they didn't just hop on some eagles and fly there.