The Eagles were a constant pain in Morgoth's evil neck; they rescued several people from his own dominion as well as attacked him personally. He must have known that the Eagles were in league with the Elves and thus surmised that the Eagles fly to and from the Elven kingdoms.
Wouldn't it have been easier for him to find Gondolin if he tracked the Eagles? If so, then why didn't he do it?
You're assuming that Morgoth knew that the Eagles lived near Gondolin, and that he could associate the coming and going of the Eagles with the location of Gondolin. He didn't and he couldn't.
That the Crissaegrim was an abode of Eagles seems to have been little secret; Of Beleriand and it's Realms refers to it as such at a time before the building of Gondolin:
Upon the left hand of Sirion lay East Beleriand, at its widest a hundred leagues from Sirion to Gelion and the borders of Ossiriand; and first, between Sirion and Mindeb, lay the empty land of Dimbar under the peaks of the Crissaegrim, abode of eagles.
But there's absolutely no evidence to connect this with Morgoth's knowledge (or lack of knowledge) of the location of Gondolin, as is shown by Of the Ruin of Beleriand and the Fall of Fingolfin (with my emphasis):
Of Nargothrond he knew indeed the name, but neither its place nor its strength; and of Gondolin he knew nothing, and the thought of Turgon troubled him the more.
It wasn't in fact until after the release of Húrin that Morgoth got his first inkling of where Goldolin lay:
Yet there were ears that heard the words that Húrin spoke, and report of all came soon to the Dark Throne in the north; and Morgoth smiled, for he knew now clearly in what region Turgon dwelt, though because of the eagles no spy of his could yet come within sight of the land behind the Encircling Mountains. This was the first evil that the freedom of Húrin achieved.
This passage also confirms that the Eagles did a pretty good job of keeping Morgoth's spies out of the area. As Thorondor himself said:
If the Eagles of Manwe were wont to err thus, then long ago, lord, your hiding would have been in vain.
To re-emphasise, an earlier passage in Of the Return of the Noldor establishes that the main purpose of the Eagles was not to assist the Elves but rather to keep watch on Morgoth and bring news to Manwe:
For Manwe to whom all birds are dear, and to whom they bring news upon Taniquetil from Middle-earth, had sent forth the race of Eagles, commanding them to dwell in the crags of the North, and to keep watch upon Morgoth; for Manwe still had pity for the exiled Elves. And the Eagles brought news of much that passed in those days to the sad ears of Manwe.
And so there is no reason for Morgoth to even think that tracking the Eagles would give him the location of Gondolin.
Gondolin and its people were protected by more than the Echoriath.
The element of Water was absolutely central to 'Ondolinde' - City of the music of stone. Ulmo was especially interested in their safety and the many 'fountains' within are mentioned over and again. Ecthelion of the Fountain was their chief captain for instance. Moreover Ulmo went out of his way to send a specific warning to Turgon about the impending fall of the city when he saved and dispatched Tuor. Moreover he had explicitly TOLD Turgon he would do this before his people left Vinyamar 400 years earlier for their second, secret home in middle-earth.
They only really began to become vulnerable when Turgon started to believe in his own strength AND Ulmo began to withdraw his influence from Beleriad after the Nirnaeth and the poisoning of Eithel Sirion.
So far as the Great Eagles go - at the very least they were extremely favoured creatures of Manwe; His mesengers with His protection on them. It is also almost certain that some of their number - if not all all - were actually Maiar of Manwe wearing the fana of birds. Therefore the degree to which Melkor could interfere with their duty or even harm them was extremely limited.
And at the end of the day, for all his unimaginable power - as perceived by Elves and Men - Melkor was as totally helpless and insignificant when compared to Eru as the First and Second children were to Melkor. Probably the difference in power was even more marked - the 'infinite' is not just simply very, very large. NOTHING happened in Ea or out of it that was against the Will of Eru. Nothing. Even the Discords were actually the working of Eru's will in order to maker better music as a result. This was something Melkor simply could not grasp.
