I seek to find research groups in my field, start a dialogue with potential supervisors, and apply for available positions. In other words, I want to check research groups' web pages mainly, not just professors' information on university websites. What is the best way to optimize my research?
It sounds like you are already in a field or have decided what field you want to be in. I started by thinking through (or looking through) the papers I had written at that point, and identified the people or groups of people I cited often, or those whose papers I enjoyed reading. Make a short list including their institutional affiliation from their papers. For some this will then be an easy step to searching out their lab group website at their university, or a link to an externally hosted site. Some may have a 'lab group' on researchgate. For those who do not, look them up in a database and see who they co-author with most often or most recently. This may help you identify their group even if they do not have a formal online presence.
In terms of starting a dialogue and applying for available positions - this depends what stage you are at and what field you are in. If you were looking to postdoc in my field, that initial conversation would include some fellowship or funding examples I was looking to apply for to come to their lab. You need to be thinking about what you are bringing them from the get go.