This is one among the most commonly reported issues. There might be a number of reasons why this happens and here are some of the prominent reasons. The following article might be of help to you in troubleshooting cart-related issues.
Item is added to cart. Clicking on the inline link doesn't take you to the cart page.
Causes & Resolutions:
Check if a menu has been created of the type J2Store->Cart and access has been set as anything apart from Public.
Check your product type. If you have a booking type product, then make sure that the cache expiration time is set to minutes higher than 0 in the booking app's settings.
Ensure that the time zone is set to the corresponding time zone as per the store profile. For example, if you have set up Denver, USA as the store address, then the corresponding time zone should be chosen in your Joomla configuration (under System->Global configuration->Server tab->Denver).
Check if the EU-privacy directive plugin is enabled under Extensions->Plugins. If so, disable this plugin and that should solve the issue.
Check for cache plugins like JSGcache, Shlib, system-page cache, system-J2page cache. Check if caching is enabled in Joomla global configuration, check if varnish caching and server-side caching is enabled. If so, disable them and then check.