Each game is stand-alone for the most part, right? I always was interested in the first two, but if it's some big trilogy that ends on a console I don't even have then I won't bother. (The prices on those games on their site is very tempting, I'll admit.)
I'm very, very into the Ar tonelico world and the lore-freak in me is screaming "No absolutely not!"
But in reality, it's more like "Yes and no." You can play them stand-alone, but some things will be a bit confusing at first.
The games are all connected using the same lore and characters. For example, terms remain the same throughout. Reyvateil, Beta Reyvateil, Third Generation, Origins, Hymnos, Servers, Viruses, the like. AT2 is different than AT1 and AT3 in almost every way, lore-wise, because of the differences in tower and server structure. For example, a character from Tower 2 appears in AT3 and is still able to sing even though Sol Cluster's Song Server is down because she is structured completely differently and her "data" comes from a different server, and her SoulSpace resides on a different server as well.
If none of what I'm saying makes sense to you, there will be times you're confused. Especially during the second half of the game.
What you really need to remember is that this is actually a science-fiction game. They use words like "Song Magic" and whatnot, but it isn't magic and the cute little girls are basically computers who run off of larger networks and servers.
I'd say AT3 is probably the worst of the bunch to start with, though, since the lore is so heavily ingrained into it. There are entire parts of the story you'll miss (For example, why items like the Heart of Gaea exist and where it comes from). But other than the important lore bits, the stories are relatively stand-alone except for some cameos.
Also, this game takes a while to get started. If you drop it within the first hour I'd suggest playing a bit more. If I only played for an hour, I'd end up with a bad taste in my mouth. I'd say the game doesn't even start until halfway through Phase 2.