Within forums, unread topics are indicated by the big dot to the left of the topic title (clicking the dot should take you to the first unread post). Topics with no new posts will have no dot.
At the forum level, forums with unread posts will have a dark green icon to the left, whereas forums with no new posts have a lighter green.
You may well find that due to the upgrade, the forum software doesn't know which posts you have/haven't read, as there is no way to carry this over from the previous software.
You can mark an entire forum as read by clicking on the icon to the left of the forum name. You can mark all posts as read across the site with the "Mark site read" link in the top right, just above the top banner.
There are a few new tools in Invision Power Board (IPB for short) to allow you to find new content. There's an "unread content" link in the top right, again just above the top banner, which works similar to the "Unread Posts" page in phpBB. You can also use the "Activity" tab on the grey tab bar to find some other options to discover content. You can even create your own custom "Activity Stream" with whichever filters you like.
(Sorry for the big images, it's just the way my laptop takes screenshots)