Tin whistle (aka penny whistle or Irish whistle) is an easy and cheap instrument for beginners. If you are new to the instrument, you don't need to invest more than $15 to get started. There are many models around this amount, but I would recommend going with a Clarke Celtic Tin Whistle in D. This is what I have and it seems to be highly regarded by others. The whistle is under $15 online. It includes a fingering chart with several basic tunes.
Here is a video of someone playing Scarborough Fair on this whistle to give you an idea of what it sounds like:
Learn Tin Whistle
While tin whistle is a fairly easy instrument, it may be a bit daunting to learn without any guidance. If you are a new player and don't have access to a teacher, you may want to pick up a book such as The Clarke Tin Whistle to help you learn. This book includes step by step instruction on how to play, reading music, and even ornamentation. It includes a CD with audio of all music in the book.