An easy way to start finding altmetric data is by downloading the Altmetric Bookmarklet from Digital Science. Digital Science is a key provider of altmetric data curated via its many platforms.
When you open an article's Digital Object Identifier (DOI) via the bookmarklet, you will see its “Altmetric Attention Score.” This score indicates different types of engagement across the open web, including news outlets, blogs, tweets, and even Reddit posts. Please note that you will need to use a DOI in order to get this score.
Another resource from Digital Science, Dimensions, offers free access to the Publications part of their database. Each article has the altmetric along with a "Dimensions Badge" that looks at the number of times a publication has been cited by other publications in their database, among other citation ratios.
Finally, check with your liaison librarian! Digital Science gives librarians access to a basic version of their Explorer database and, depending on your needs, your liaison may be able to gather a little more information.