Ar Pharazon blew either horns or trumpets (I always get them mixed up
) when he went to war against Valinor.
Fingolfin blew his horns/trumpets when he started the march over Helcaraxe, and when he came in sight of Thangorodrim. He also blew his horn/trumpet when he challenged Morgoth.
Horns/trumpets were used in almost all battles. The one that specifically jumps to my mind is Helm's horn, which was blown at dawn at the battle in Helm's Deep. Also, in my head Nirnaeth Arnoediad is linked with trumpets/horns, because Turgon blows them before he goes to battle to help Fingon, and Fingon hears them, etc (we know the story). In the War of Wrath, I think it says something like "the sound of the trupets of Eonwe filled the land". And when the Rohirrim reach Minas Tirith they blow horns.
I find it interesting how Boromir wants to begin the mission of the Fellowship with blowing his horn...
At the moment I can think of two other musical instruments that are used: drums and harps. Drums are used by the Druedain to send signals, and I
think drums were used by the Mordor army when they attacked Minas Tirith. I
think the Rivendell menestrels accompany themselves on harps, but I don't know for certain. Finrod plays a harp when he first met Beor and his people; Thorin plays a harp in Bag-End.
And speaking of Thorin, his Co. has a few other instruments that I can't remember. I think there were fiddles and flutes, but I'm not sure.