Pope Benedict XVI said in 2005:
"I would like in particular to recall and recommend the ancient tradition of Lectio divina: the diligent reading of Sacred Scripture accompanied by prayer brings about that intimate dialogue in which the person reading hears God who is speaking, and in praying, responds to him with trusting openness of heart (cf. Dei Verbum, n. 25). If it is effectively promoted, this practice will bring to the Church - I am convinced of it - a new spiritual springtime."
The FishEaters website has the best explanation of how to do Lectio Divina that I have come across. It is excellent and gives you a good history of this method of prayer as well. Most of the sites that I have seen that write about Lectio Divina try to incorporate it with Eastern forms of prayer, which are not Christian, so I was happy to find this one.
While the author of this prayer guide is a big proponent of the Douay-Rheims translation of the bible, which is the Catholic bible used until 1970, you should feel comfortable to use any Catholic bible you have. In fact, it is a good idea to own several translations to compare passages. Beside the Douay-Rheims translation, there are two other excellent translations you might want to explore: the Revised Standard Version - Catholic Edition (RSV-CE, commonly known as the Ignatius Bible) and also the Jerusalem Bible.
The RSV-CE is the Bible used by some of the best and most faithful Catholic bible scholars, and is one of the most accurate translations in English. There is a new Second Edition that has a bit more modern English, as well.
A good Catholic bible commentary was recommended to help understand passages for prayer. An excellent commentary you should check out is the Navarre Bible. It is the best commentary for laymen that I know of. It comes in individual volumes of the books of the bible or in clusters of books.
Like all new things, Lectio Divina, takes a little bit of practice, but in a short time each stage will flow easily from one to another without much thought. Praying in this format will bring great peace and blessings to you and help you in your quest for closeness with God.
Here are a couple of other links that may be useful as well:
Lectio Divina and the Practice of Teresian Prayer
What is Lectio Divina?
UPDATE: Here is a great article from the National Catholic Register about using modern technology to aid us in Lectio Divina - Wired For the Word of God: Lectio Divina in the Digital Age
and a beautiful article from a Carmelite Prior General about the Fundamental Elements of Carmelite Spirituality in which Contemplation and Lectio Divina play a major part.