A good place to start is Patrick Dunleavy's Twitter guide for academics.
You don't need a Twitter account to search Twitter. The simple search facility is here. The advanced search facility is here.
A simple example. If you search for school attendance, up will pop a list of tweets. You can check out who tweeted (click on the name of the tweeter). That will give you information about the person or organisation, number of followers, who they are following, number of tweets etc.
You need to be ruthless in assessing whether or not you want to include a person in your tracking which is easily done with a twitter account or you may opt for a Google alert.
There is useful advice about using twitter in academic practice at the LSE impact blog, for example:
Academic tweeting: using Twitter for research projects
There will always be studies of use and impact on practices. Here are a couple of cases:
Veletsianos, G. (2012). Higher education scholars' participation and practices on Twitter. Journal of Computer Assisted Learning, 28(4), 336-349. Available here.
Veletsianos, G., & Kimmons, R. (2016). Scholars in an increasingly open and digital world: How do education professors and students use Twitter? The Internet and Higher Education, 30, 1-10. Available here.
Twitter can also be a good source of poking fun at many of the practices of academics across most disciplines. Andy Tattersall offers a few suggestions.
A useful piece by Mark Carrigan Support, engagement, visibility and personalised news: Twitter has a lot to offer academics if we look past its image problem
Allan Johnson shares how he uses twitter for curated academic content. The hastag #academichipster is also worth a scan for a bit of mirth.
A search for twitter on the LSE impact blog site will pull up a wide range of articles that scope the current debates about the use of this social media platform
A post in 2019 from Amanda Heffernan and Rachel Buchanan opens with this point:
Twitter is the social media of choice for many academics. At least one in forty academics in an institution is on twitter, contributing to the 4.2 million tweets about education every day. If you are involved in education in any way it is probably a good idea to get on there and see what is happening.