Mantras are very powerful prayers that seem to draw the very essence of the divine to you as you recite (chant) them. Used properly, and with the correct intent, they are able to bring about significant and lasting transformation to your live both on a physical and spiritual level. As well as improve particular areas in your life that need working on at present.
The best way to work through them is to simply choose whichever mantra feels right to you at a given time and then recite it with true intent for the next 40 days before moving on to another one. The mantra of your choosing in normally recited 108 times in a sitting but I personally feel you should chant it for as long as you feel is right for your own soul and no more. In other words, do what you feel is right for you.
When the time is right the mantra will start to feel like it is a part of you and you might then find your mantra repeating itself within your mind without any conscious instigation on your part. Repeat the mantra whenever you have time throughout your day but I especially find it works best recited first thing in the morning and often feel that when I neglect to do so things simply don't go as smoothly as when I do make time to chant. Try where possible, to make it part of your very essence.
A mantra must connect you with the sacred and I try to imagine it coming from my heart as I speak it out loud. A list of some useful mantras is provided below. Simply select the one that most resonates with you at this time and work on that to begin with. My personal favorite is Om Mani Padme Hum so I would recommend starting with that to anyone that doesn't feel a particular mantra calling out to them at this time.