It depends a bit on whether you are using the regular web version on a computer or other device or whether you are using the tapatalk app on a phone or tablet. I don't use the app, but on the full view web version you will see a button that says post reply at the top and bottom of each page of a thread.
Also from the full web view if you click on UserCP along the top menu bar you will find a section that says "subscribed threads" and in that section a link for "list subscriptions." When you click on list subscriptions you will be able to select frequency options for notifications of new posts in threads you are subscribed to.
If you want to stay logged and walk away without your inactivity logging you out then check remember me when you log in.
Also from the full web view if you click on UserCP along the top menu bar you will find a section that says "subscribed threads" and in that section a link for "list subscriptions." When you click on list subscriptions you will be able to select frequency options for notifications of new posts in threads you are subscribed to.
If you want to stay logged and walk away without your inactivity logging you out then check remember me when you log in.