Some publishers provide altmetrics on their websites to authors and the public. PLOS has been a leader in providing altmetrics. Each article they publish has a "Metrics" section which provides a wealth of information. Click the image below to see an example.
Many repositories such as the Iowa State Digital Repository track downloads and views of each item. These statistics are separate from the statistics gathered by publishers, so if you are archiving your work in a repository, make sure you take these into account!
If you created a website for your research project, you can collect statistics about who is using it. Google Analytics is a free platform that keeps track of this information and provides reports that include which state/country your users are from and which pages/items are being clicked and used.
If you are using a service like Wordpress, they may already be collecting this information for you!
Another place you can gather altmetrics is through online social interactions.
Retweets, likes, and shares are a way to track the interest level of a research project or article. Comments on blogs, reviews, and articles are another place you can look.
Another source of altmetrics are social bookmarking and reference management tools. There's been some research that shows that the number of times an article is "saved" by users may correlate to later citations (Altmetrics in the Wild: Using Social Media to Explore Scholarly Impact, 2012). Some citation management tools such as Mendeley provide readership metrics. Note that due to changes in recent years, social bookmarking is considered by some to be an outdated method for increasing the Google page rank for publications.