Where to start if you want to become more mindful?
Although meditation is not required to be mindful, it is a good tool to practice our awareness of the present moment. When we dedicate a fixed amount of time at a regular interval to our practice, our mind will be more willing to go along. After all, when the meditation is over, it can go back to doing what it wants: thinking and analysing. However, especially for inexperienced practitioners, it can be hard to practice on your own. Therefore it will be really helpful to find a group of people to practice together.
Nowadays there are many mindfulness trainings you can follow. I can recommend trainings based on the 8 week MBSR (mindfulness based stress reduction) or MBCT (mindfulness based cognitive therapy) programs. These trainings are good for learning the basics of mindfulness. There is one downside however, it stops after 8 weeks. I highly recommend finding a training that offers meditation sessions after these 8 weeks. Or find a meditation group elsewhere to continue your practice.
Mindfulness is a bit like running. Everyone can do it, but in order to get lasting results, you need to practice regularly. And support in the form of a trainer and a group of peers can help greatly.
If you would like to hear some meditations or read a book first, then here is an interesting website: http://franticworld.com. Have a look at the resources page for free meditations and recommended books.