The DREAM Lab supports a number of tools for acquiring and analyzing social media data. Feel free to schedule a consultation if these guides do not answer your questions.
A browser-based tool for harvesting and analyzing Tweets, as well as other social media sources such as Reddit, WordPress.com blogs, and select public Facebook posts. Request access to BrandWatch
A Free and Open Source Python script that uses Twitter's public API to harvest Tweets and their associated metadata
Text Data Mining
Social media data analysis can be considered a type of text data mining. The Library has a variety of other sources of texts that can be mined, and API's that can be used to compile your own full-text datasets.
There are also online sources of pre-existing datasets: