The way it went for me was, I read Mindfulness in Plain English and it's pretty good, but it's not like the best thing ever, so don't expect that. It is a very good introduction. It resonated with me as an introvert and a person full of fear and anxiety, but it's for everyone.
The activity of mindfulness is very simple (deceptively): "be aware of what is happening, and don't get lost in thought"——so the book (and much of mindfulness teaching) can seem repetitive. It's often like, "what if you feel pain? Be mindful of the pain. What if you feel bored and annoyed? Examine what it is like to feel bored and annoyed. What if I think this is stupid? Be mindful of what it's like to think mindfulness is stupid." etc.
I think the book is available as a pdf, or you know, where you get books.
After the first time I read the book, I meditated off and on for a year, but not consistently. Then one day I decided to re-read the book and also started listening to podcasts by this guy named Gil Fronsdal (recommended by Dan Benjamin), and after listening to Introduction to Mindfulness Meditation, Part 1, I pretty much stuck with it every day since.
Here are four of my favorite podcasts from when I was starting out. Start with Introduction 1 (direct link to MP3), and then I liked Introduction 5, then Concentration Part 2, and then Meditation as a Mirror. You can find many more here.
There are other good teachers you might like more than Gil Fronsdal, though he has a genial, Mr. Rogers-like way about him I like a lot. These are all on Itunes also——search for "audio dharma."
The important thing is to actually meditate, so try to do it along with them in the "Introduction to" podcasts. On my own I started out doing it daily 10 minutes, then 20 minutes, and now I do 25 minutes, using a timer. Some days it goes well and I get concentrated and some days I sit and just think about stuff and try to relax. It's one of those things that can sometimes seem like a chore, but you never regret doing it. It gets better the more you do it, like exercising any skill.
There are different meditation techniques besides mindfulness, so you if you want to try something else, or keep it fresh, there is, for instance, "concentration," which I guess is more like focusing and moving into altered states of mind, and there's "lovingkindness" (where you sit and think generous thoughts about everything, I think?) and there's of course Zen, which is Zen. I don't know that much about any of this, so I might not have this right.
There are guided meditations which are really helpful, where they walk you through it. When I was starting out I sat through some guided meditations which were amazing, and I thought to myself, "I'm going to keep meditating for the rest of my life, because this is pretty great." I highly recommend doing guided mediations because you learn tricks for later, when it's just you on your own. There are some here.
The video mentioned in the third panel above is here.
For further views (by Westerners) of Buddhist approaches and teachings, and how it is a secular and practical way of life, I found the one on Blasphemy pretty interesting, made around the time of all the Benghazi madness, and also this one on The Importance of Questions by Geoffrey DeGraff is a kind of basic introduction to what was the deal with the Buddha.