If you are starting lessons it is good to check with your instructor before investing in a flute or whistle. I'm happy to consult with you before your flute purchase if you are coming for lessons or classes.
Often a student will begin their lessons using a tin whistle. These are inexpensive and easy to come by. Most music shops and online locations carry tin whistles in D for under $15. Other keys are often available but in order to play at the same pitch as a flute with others, use a small "high" D whistle.
A good starter flute can be a "composite" flute made of delrin or polymer. The best of these flutes are machined just like a wooden flute but with less expensive materials. Most flute players keep their composite flute forever, to leave out on a table at home or to use in inhospitable climates, because they are virtually unbreakable.
A wooden flute without keys is the simplest choice. You will not be taught any tunes which need flute keys. Most traditional Irish music does not require a keyed flute.
An Irish flute with 'all-fingers-on' will have you covering six holes. The standard note which this sounds is D. (this is called a 'D flute'.) That's the kind you want.
Local sessions with flute players, Comhaltas (pronounced "COAL-tus") regional events, and the Catskills Irish Arts Week are all good places to try out a wooden flute. Some of these events include instrument-maker showcases, where you can try a new flute, meet the maker, and order an instrument if you wish.
In a session, a polite inquiry to a flute player could end with an offering to try out the feel of their flute.
Tin Whistles
Some great low-cost whistles are available from Jerry Freeman at his ebay shop.
Higher-end Irish whistles are made by John Sindt, Michael Copeland, Michael Burke, and others. (contact me to connect with John Sindt)
Irish Flutes
Irish flutes are not usually found in a local music shop, but there are many ways to purchase one.
The Guide to the Irish Flute prepared by Brad Hurley will give you a very good idea of what an Irish flute is and who makes them.
John Gallagher (of Elkins, West Virginia)
There are many more makers listed on the flute makers directory. All these instruments are hand-made and often have a waiting list.
Rob Forbes, Paddy Ward, and Michael Cronnolly make great composite flutes.
Chiff and Fipple is a lively community with many flute players buying and selling flutes. Usually a flute player there will allow a week or two grace period where a buyer can try the flute and return it if it's unsuitable. Many flute makers have a similar policy.
Blayne Chastain runs the online IrishFluteStore.com and is a great resource, as well as a cracking good flute player.
You can buy flutes on Ebay, but beware of imitation flutes which are there in abundance.