I ask this because I was wondering, since Morgoth came across them first, they had to have had some sort of language in order to communicate with him - or is that not specified?
Mannish (mainly Taliska, giving rise to Adunaic and Westron; also Haladin, giving rise to Dunlendish); also Sindarin during their time in Beleriand.
From Tolkien Gateway (that rare thing - a reliable wiki!):
In Beleriand most of the Edain adopted the Sindarin tongue, although their native language was Mannish with elements from Avarin after their first contact with the Dark Elves. The language of the survivors of the First and Third Houses was the ancestor of Adunaic; the language of the Second House was unrelated to them.
More specifically, the Edain of the House of Bëor and the House of Hador spoke a language called Taliska. By the time the two tribes reached Beleriand, they already spoke different dialects; in Beleriand, both began to speak Sindarin (Elvish) instead. The language of the folk of Hador was more pure and they felt that the dialect of the folk of Bëor had alien elements.
The language of the House of Bëor eventually was almost extinct, both because of the adoption of Sindarin and because the House was mostly annihilated after the Dagor Bragollach. But the House of Hador did not wholly abandon their language: it survived the War of Wrath and was still spoken when the Edain migrated to Elenna, later giving rise to the Adûnaic (Numenorean) language, which in turn gave rise to Westron (the Common Speech).
The third Edain House, the House of Haleth, spoke an unrelated tongue, known as Haladin. We can see in this excellent schematic that this tongue later gave rise to Dunlendish, the language spoken by Ghan-buri-Ghan and the men of the woods who appear in The Return of the King.
All of this can be sourced to History of Middle-Earth, volume XII: The Peoples of Middle-Earth, section "Of Dwarves and Men".