Some might like a video to bring the tales alive, others would want to read either in a book or on a website. Yet others prefer to close the eyes and listen. Or perhaps it depends on your mood. Below is a choice list to find what suits you best.

Videos by Cyb the chuckling monk offer one of the gentlest and most fun introductions.
The FREE sample videos have the story for listening, with beautifully chosen photos, artwork, and music. Though beautiful this is not a complete set of the Four Branches, nor is it a close telling from the original. But lots of fun. www.valleystream.co.uk/products.htm

Listen to readings by Colin Jones. The complete First Branch is FREE on the website with an atmospheric musical background. Or there is the complete set of the Four Branches download/DVD to buy. This is the classic Charlotte Guest translation. http://themabinogion.com/album/mabinogion-the-four-branches

If you only want to buy one book you you could choose one of two recent classics.
Sioned Davies' ttranslation: The Mabinogion (2007). A handy small sized book with useful notes, and a clear, friendly, storytelling style. This book has all the other stories of The Mabinogion as well as The Mabinogi.
John Bollard's translation: Legend and Landscape of Wales: The Mabinogi (2006). A gorgeous book in friendly style, packed with magical photographs of the actual sites where the events happened by Anthony Griffiths. The other mediaeval Welsh stories are available in separate volumes.

For FREE online reading go to Will Parker’s generous website. www.mabinogi.net/translations.htm
If you prefer to read on paper you can copy and paste the four (longish) pages. Lots of useful notes if you want to explore further, and based on reputable sources. Will Parker also has lots of thoughtful articles as intermediate studies.

The pioneering publisher and translator who made the Mabinogi available to us in the modern world, was Charlotte Guest. Check out her extraordinary life here, and pay a courtesy visit to her book which is freely available online. www.sacred-texts.com/neu/celt/mab/index.htm Lovely Victorian illustrations by Samuel Williams the foremost engraver of the period.