I think the easiest way to try mindfulness meditation out is the headspace app, but there are a lot of different styles if that doesn't appeal to you. Most mindfulness-oriented meditation taught in the west today won't demand a set of beliefs and compatible with a secular rationalist perspective on the world.
There is definitely technique; it's quite different from just daydreaming or following every thought that enters your mind. You will typically have something to focus on, usually the breath is used to start, but it varies by tradition. The idea is to have something neutral that you can pay attention to so that you can begin to observe the thoughts that arise in your mind without reacting to them. There's a popular free book available online in various formats called Mindfulness in Plain English that lays out the basics of Vipassana style meditation, if you want a decent primer. The retreat I mentioned was offered by dhamma.org aka the Goenka school.
Feel free to message me if you have any questions I might be able to answer.