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 provide authors with a dashboard (example below) that show statistics about their works in the repository. 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!
Another place you can gather altmetrics is through online social interactions. Not all platforms are open, but Twitter for example, has an open API which lets other websites connect to it and harvest statistics.
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).