There's no single "the meditation", just like there's no single "the exercise". There are many exercises and many meditations. What you do in a meditation is tied to what you are trying to achieve, and that's where the right/wrong judgment comes from. Consequently, if you are not trying to achieve anything in particular - then there's nothing you could be doing right or wrong :-P
In your case, you are just starting to explore. As a beginner, there are two ways you can go from here:
- You can keep exploring. Don't fight with your thoughts but don't get caught up in thoughts either. If you do get carried away, gently come back to the present moment. In this open-style meditation anything goes - including thoughts, memories, the environment, all kinds of mental trips - as long as you keep coming back. It's like the laundry machine, it keeps cycling but always stays in place. Pay attention to all the ebbs and flows, explore your mental/emotional/somatic continuum, and give your introspection some time to mature. Some serious time, at least six months to a year. You can't force the natural progress, it has to take time. Your goal is to learn the landscape through random roaming.
- Alternatively, you can take up the breath meditation proper. In this one, your goal is much more narrowly defined. Your goal is to achieve pinpoint focus on the microscopic nuances of how each of your in-breaths and out-breaths feels second-by-second, and through that to access your feeling mind. This requires tremendous level of concentration, exclusively on the breath (and on the feeling mind) not engaging in thinking, reminiscing, observing the environment, meta-analysis, or anything else. Your eventual goal with this meditation, once you learn to access the feeling mind, is to start untangling the emotional tangles, untying the knots and straightening out the old wrinkles.
Eventually, with both types of practice, you should arrive at this open, relaxed, transparent, lucid, and even psychosomatic continuum, like looking down at a very quiet koi pond.
Anecdotally, for many people the specificity of breath meditation seems like a more comfortable more deliberate framework than the open-ended exploration-style meditation. Then again, since our minds are pretty restless, most people end up cycling in and out of their thoughts during the breath meditation, blaming themselves for getting distracted, ending up neither here nor there.
Right now you are kinda trying to do both meditations at once, worrying whether you should focus on the breath or allow your mind to settle and explore naturally.
I recommend to make up your mind and pick either one meditation style or the other, and commit to your choice for at least six months, better a year. Once you get some experience with either one, you'll have enough insight into the functioning of your mind to make a meaningful next step.