What happened to Morgoth's original fortress after he was imprisoned? Was it completely demolished by the Valar? Did Morgoth just abandon it for Angband? What happened to it when Morgoth returned to Beleriand in the First Age?
It was destroyed
But not completely; the fortress itself was destroyed, so it was no longer a viable base of operations, but the tunnels and catacombs beneath were too extensive for the Valar to fully destroy or explore, and some of Melkor's minions hid there:
Long and grievous was the siege of Utumno, and many battles were fought before its gates of which naught but the rumour is known to the Elves. [...] The lands of the far north were all made desolate in those days; for there Utumno was delved exceeding deep, and its pits were filled with fires and with great hosts of the servants of Melkor.
But at the last the gates of Utumno were broken and the halls unroofed, and Melkor took refuge in the uttermost pit. Then Tulkas stood forth as champion of the Valar and wrestled with him, and cast him upon his face; and he was bound with the chain Angainor that Aulë had wrought, and led captive; and the world had peace for a long age.
Nonetheless the Valar did not discover all the mighty vaults and caverns hidden with deceit far under the fortresses of Angband and Utumno. Many evil things still lingered there, and others were dispersed and fled into the dark and roamed in the waste places of the world, awaiting a more evil hour; and Sauron they did not find.
The Silmarillion III Quenta Silmarillion Chapter 3: "Of the Coming of the Elves and the Captivity of Melkor"
It still existed and Morgoth's plan was to return there after destroying the elves.
In a c.1960 work titled "The Awaking of the Quendi", Tolkien talks a bit about the relationship between Utumno and Angband. He concludes that these were different places, with Utumno being Morgoth's main seat of power, and Angband being only his western outpost, to guard the shores and to produce a lot of smoke. Morgoth spent the bulk of the Silmarillion in Angband, but he saw this as only a temporary thing that he was doing while at war with the Noldor, and he planned to return to Utumno at first opportunity.
As soon as he discovered the Quendi (if not indeed far sooner, and well before the time of their awaking, which Melkor guessed more shrewdly than the Valar) Melkor constructed Angband. One of its chief functions was not only to defend the Western Shores, but to shroud them. The prime function of (originally volcanic) Thangorodrim was to produce smokes, vapours, and darkness. All the Northwest shores were covered and the Sun largely excluded for hundreds of years before Melkor was made captive. Sauron had a chief part in this; and when the Valar at last came to Middle-earth he (under Melkor’s orders) made a strong feint of resistance, while Melkor retreated and gathered nearly all his forces in Utumno. (Thus passage of the Quendi was made feasible.) Angband was in the event very largely destroyed – though the Valar, passing on to Utumno, which was apparently the real centre of Melkor’s power – made no attempt to demolish it completely. But when Melkor feigned submission to Manwë, Sauron was ordered to reconstruct it (as secretly as possible: therefore largely in extending its underground mansions) against Melkor’s escape and return. There were no more fumes until Melkor returned: but when he did in 1495, Angband was almost ready. Melkor then made it his chief seat of power, for strategic reasons, and because of the coming of the Eldar. Had he been successful he would perhaps have returned to Utumno, but not until the Eldar were vanquished or destroyed.
The Nature of Middle-earth - "The Awaking of the Quendi"