I have been interested in mindfulness and meditation for a long time (>10 years). Of course none of this was practiced well and with any consistency when I was drinking heavily, but got back into them in early sobriety.
When I first got interested in these things, I was struggling on my own so decided to look for some guidance. That led me to investigate a few different teachers and techniques, but it never truly clicked and kicked in until I found what and who was a natural fit for me. I found an ad for a meditation retreat online that seemed very intriguing, then I researched the source and the approach, and I got totally fascinated even just reading about it. It's called vipassana meditation, and my teacher is Alan Wallace. You can read about his work online. I signed up for a one-week retreat ran by him, and from the first day I knew I was at the right place. It was an amazing experience and I also met many like-minded people at different levels of their practice. Alan Wallace himself is the kind of mentor that I would happily have for anything I am interested in, his entire being and approach is wonderfully compatible with me.
An experience I had at start though was that I got intrigued by and wanted to jump into a very high level of it without having knowledge about and experience with the basics. Like the elephant in the porcelain store. My first retreat was also far from a beginner's course and while I enjoyed it very much, I only appreciate later looking backwards how I needed to start in the beginning to even comprehend accurately what it was about and how to use it for my benefit.
What I would suggest to anyone new interested in mindfulness and meditation is to get yourself informed and try different approaches. Some people prefer to do it on their own, I personally benefited a lot from having a guide and some structured learning experience.
A couple good books on the subject that are related to addiction:
The Mindful Path to Addiction Recovery by Lawrence A. Peltz
The Mindfulness Workbook for Addiction by Rebecca E. Williams
Other relevant authors I like are Pema Chodron and Eckhart Tolle (the author of The Power of Now that zjw mentioned).
Perhaps Refuge Recovery might also be interesting to you; they also have meetings in many parts of the world. Also, as mentioned above, there are many therapists that use these tools in their treatment approach. If you are interested in something perhaps more physically active but with essential elements of mindfulness, I find yoga very helpful.
I am also prone to anxiety and have had it at variable levels throughout my life. I choose not to take medication and manage it with tools like these, plus exercise and therapy. These things have worked wonders for me in sobriety.