The "base" instrument in Celtic music is the human voice, and this confirms the fact that folk music stretches naturally towards melody and not towards harmony (1). The musical instruments nearer (from a expressive point of view) to the human voice are the violin (fiddle), the flute, the tin whistle and the pipes (bagpipes in Scotland, uilleann pipes in Ireland). It's impossible to play chords on the flute or the tin whistle, and even if it's possible to play them on the fiddle, they are seldom used this way in Celtic music. The only instrument from the Celtic musical tradition where chords are possible is the Celtic harp; so important is this instrument, that over the centuries the same has become the symbol of Ireland.
The pipes constitute a particular case, because they have a bass system (drone) and the possibility (limited) of playing some chords, but they are used only occasionally, and more for harmonic than rhythmic effects, so it is reasonable to also classify the pipes as a melodic more than harmonic instrument (2). Other instruments used in Celtic music are the button-accordion, the concertina and (very rarely) the harmonica. On these instruments it's possible to play chords but (as for the pipes) they are used more for rhythmic emphasis than for the accompaniment. The typical percussive instrument is the bodhrÓn (3).
The Internet has many web sites dedicated to the use of these instruments (where to acquire them, technical advice, maintenance, etc). Here are the more interesting ones, divided for single instrument:
Mandolin - cittern - bouzouki
Concertina - Accordion - Harmonica
Uillean pipes & Bagpipes