Not being able to read music isn't a liability when it comes to irish traditional music. Most folks will say that it's really sort of a liability TO be able to read music. These tunes have been passed on by ear for generations. Some of the greatest players (of any instruments in the tradition) can't read a note - particularly from the older generation. I spent a week last year with
Kevin Crawford of Lunasa and was surprised to find out that he doesn't read a note.
My flute & whistle teacher couldn't read music but made himself learn when he started teaching here in the US. Listen to as much of the music as you can stand. That's one of the best things you can do to learn - particularly once you find a player whose style of playing you like.
Another thing to do is check out
The Session and see if there's an irish music session near where you live. There'll probably be a whistle player there who can help you get going.
There are a bunch of tutorials out there too. I got started playing using
Cathal McConnell's Tutorial from Homespun Tapes. There's also a fella on youtube who's a good player who has a of tutorial as well -
Ryan Duns.
Here's the start of his lessons.
Some other links that'll be good for you are
Chiff and Fipple - they have a good forum there (just watch out for the obamunists over there) and
Whistle This. Whistle this is kind of dead these days, but it does have tablatures for the tunes which you might find helpful.
If your time and wallet allow, there are some great irish music camps over the summer at
the Augusta Heritage Center in Elkins, WVa and at the
Swannanoa Gathering near Asheville, NC (this is where I met Kevin). They're kind of pricey, but they're well worth it.
Best thing you can do though is listen. Especially if you want to learn a new tune. When I find a tune I want to learn, I listen to it over and again until I can whistle/hum it. John tells us "you can't play a tune until you know it", and he's spot-on there.
Your Oak & Walton will get you along a good way in learning to play.
If you want to check out some whistle playing that'll blow your mind (although it isn't necessarily all that traditional), check out the band
Flook.
ETA: This'll be a good spot to plug where I go for lessons when I can:
The Riley School of Irish Music in Cincinnati, Ohio. There are several years worth of tunes on the site that you can download (both slow versions and more "up to speed" versions).