Whereas Facebook and Twitter have powerful built-in tools for analytics and there are a pleothra of free third-party services, Instagram lacks robust internal insights and there are few outside services. However, there are a few free options that are worth investigating if you are trying to better gauge the efficacy of your Instagram content strategy.
Simply Measured is a powerful analytics suite which is used by some of the leading brands on social media. Although their full suite can be a little pricey, Simply Measured provides several free reports for your use including a Instagram User Analysis. In order to access the reports you will have to provide some personal/business information as well as connect with an Instagram account. It is good to note that you do not have to connect the same Instagram account that you are running a report on - this is helpful to run competitor reports and new client recon. Helpful metrics include engagement per content type (photo / video), top performing posts, top day for engagement, and most frequent commenters.
Socialbakers provides a sleek interface to review your post popular posts by likes and comments as well as your most used hashtags. They also provide the helpful metric of what accounts you have liked the most content from. Another helpful graph is one sowing the breakout of posts by month which can be viewed for all years. Finally, Socialbakers shows you your most used filters which can assist in ensuring you are putting forth a consistent brand image across posts.
Iconosquare offers a free 7-day trial of their analytics platform, which can be helpful for one-time deep-dive reporting. Provided are several helpful metrics such as love rate, talk rate and spread rate which are provided for both your most recent post and an average of your last 15 posts. Iconosquare's Community reporting tools are helpful in that they show how many followers follow you back vs. those who don't follow you back.