I had a similar kind of problem actually. There are a couple of things you can do to get Instagram to start letting you follow people again:
1. Close the app and then open it again.
If you're using an iPhone with iOS7 then you need to do this to close Instagram.
Open the app, double click the circle button, it will show you a bunch of windows of apps that are active. Swipe the Instagram app towards the top of the screen and it should disappear.
Try opening Instagram and follow people again.
It should work (it worked for me), but if it doesn't - there is still something you can do:
2 - Uninstall the app and reinstall it.
Yes, it's a pain and takes a few minutes.. But this is like the 'master reset' option.
If this still doesn't work, then this means there's nothing technically wrong with the app, and your account may have been blocked. In which case, try contacting Instagram support here:
help.instagram.com/