Before you dive into Buddhist books you should start with the right attitude. The following metaphor may help (please forgive any generalisations): there are two pills, a sugar pill and an Anadin pill. The sugar pill has absolutely no inherent power to cure headaches. However, if you believe strongly enough that the sugar pill will cure your headache then your pain will actually disappear. Compare this to the Anadin pill which will cure your headache whether you believe in it or not. A lot of what you will read in these Buddhist books can be divided into sugar pills and Anadin pills. My belief is that some Buddhist rituals are actually sugar pills, they have no intrinsic worth but if you believe wholeheartedly in their benefits they actually will help you. Meditation, on the other hand, is like an Anadin pill, it has inherent value, it will help you even without your belief in it.
So tread softly, pick out the Anadins and feel free to leave the sugar pills behind if you don't have faith in them. There is no need to accept everything written in these books (I certainly don't) and try not to dismiss all of Buddhism just because many of the ideas are unbelievable to you. Buddha said not to believe anything he said just out of respect for him, he told his followers to test his words as carefully as a jeweller would test gold.
I am hoping that the order in which I've listed the books below is the most beneficial order in which they can be read. It starts with The Road Less Travelled which introduces the importance of spiritual growth. Then, if you can't actually meet inspiring Buddhist monks in the flesh, maybe you can be inspired by the writings of Thich Nhat Hanh whose lovely personality and character shines through in his books. The ancient Dhammapada is a collection of the sayings of Buddha and is short, full of wisdom, free of difficult concepts and easy to digest.
Mindfulness meditation is, for me, the most significant aspect of Buddhist teaching and after getting a taste for its importance from Thich Nhat Hanh I think it's a good idea to learn more about how to apply the technique from some Western literature, i.e. the Insight meditation books and the simple and practical How to Meditate.
After that it will be time to delve into the fuller world of Buddhism. Many of the concepts will be strange, maybe unbelievable, to you I'm sure. I used to think reincarnation was complete rubbish. Today I can't say I believe in it but I now recognise a lot of sense in the idea and I certainly believe in the possibility of its existence. Couple this with the fact that Christ mentioned very little (and arguably nothing) of significance on the subject of what happens after death and there's definitely some leeway for even Christians to believe in reincarnation. So switch on that open mind of yours! :-)
I haven't been able to read most of these books yet though I've read bits of quite a few. Many were recommended by the Tushita teachers.
The Dhammapada: "Those who delight in truth sleep peacefully, with clear minds. The wise always take pleasure in truth expounded by the noble." Please enjoy the expounding of the Buddha ...
M. Scott Peck - The Road Less Travelled
Okay, this is not a Buddhist text but it was inspired in part by Buddhism and it is a great book about the path to spiritual growth. A glut of self-help books seem to pour out of the U.S., most of them cash-in-quick crap that try to fool people into believing pure happiness and wealth is only 7 simple steps away, but this one is a diamond. Read it and if it doesn't do anything for you then you can skip the rest.
Thich Nhat Hanh - Being Peace
Written by a Vietnamese monk who was once nominated for the Nobel Peace Prize by Martin Luther King, this is simple to read, a very inspirational book on mindfulness. Thich is, along with the Dalai Lama, probably the greatest Buddhist teacher living today and this book is a must. If you like it please try his Living Buddha, Living Christ or The Miracle of Mindfulness, Transformation & Healing, Peace Is Every Step and The Sun Is My Heart.
Buddha - The Dhammapada, Translated by Thomas Cleary
There are numerous translations of the sayings of Buddha floating around but I found this one to be very good because of the helpful comments scattered through the text, illuminating the meaning of various verses. The sayings are basically a set of wise proverbs on how we should conduct ourselves and are easy to read.
Joseph Goldstein - The Experience of Insight
Joseph Goldstein - Insight Meditation
Joseph Goldstein, Jack Kornfield - Seeking the Heart of Wisdom
Meditation is the real Buddhist key to happiness and these books are on insight (vipassana) meditation, a practice developed by two Americans who trained with numerous Eastern meditation masters (including Goenkaji from the Vipassana course I did in Nepal) and have blended their knowledge into a course that's tailored to Western minds. Highly recommended. Read at least one.
Kathleen McDonald - How to Meditate
Very simple and clear, a good intoduction to a variety of different meditation techniques.
Sogyal Rinpoche - The Tibetan Book of Living and Dying
This one's a bestseller by the best known Tibetan author after the Dalai Lama, beautifully written and highly recommended for all those interested in healing or learning more about Buddhist ideas of death, dying and the mind. Visit Sogyal's organisation RIGPA on the Net, they have a retreat centre in southern Ireland.
Thubten Chodron - Open Heart, Clear Mind
Great introductions to Buddhism are hard to find but this one, by an American nun, is pretty good. Has a good section on dealing with emotions.
The Dalai Lama - The Way to Freedom
If you'd like to read an introduction by His Holiness then this one was written especially for Westerners. Basically anything by him is recommended, e.g. Healing Anger, A Policy of Kindness, The Good Heart, Healing Emotions, Beyond Dogma and his autobiography Freedom in Exile.
Shunryu Suzuki - Zen Mind, Beginner's Mind
Written by a great Zen master, this book is a classic guide to Zen and meditation.