It seems from reading the other discussions in this lesson that "dia duit" and "slán" have literal translations that relate to a slightly different meaning ("God be with you" and "be safe" according to @xounds, respectively), even if their typical usage sheds those connotations. Does fáilte do something similar?
This may help http://en.wiktionary.org/wiki/f%C3%A1ilte#Etymology
It used to mean delight or joy but now its slightly less specal :)
The 't' is followed by an 'e' and is therefore palatalized. That is, it's pronounced rather like the 't' in the non-American pronunciation of 'tune' (tyoon not toon).
It shouldn't be an 'sh' sound. You might be misperceiving the palatalized t (very easy when you're not sure what you're hearing), though it could also be that someone is pronoucing it oddly. Bear in mind that most Irish people are not native speakers of Irish.
I am learning Irish whilst living in Ireland and nearly EVERYTHING is pronounced differently. Here they would say fal (as in falcon) and cha for failte. If you are going to learn it, it's best to learn the pronunciation used on this programmed unless you are learning specifically to a dialect.
Question: in 'Tá fáilta romhat', the 't' in 'tá is palatal while the 't' in 'fáilte' is dental?
...someone somewhere else published this link to a pronunciation guide:
Granted, there's a LOT in this guide, but it has made me start listening more closely to specific sounds.