You can only see activity from people you're following on Instagram and to do that you click on the following option at the top of the screen on Instagram.
If the person on Instagram is following more than 1,000 people then the following tab will not work and you won't be able to see any activity from the other persons Instagram account.
Also if the person on Instagram has there account set to private you won't be able to see who they're following or any of there other activity.