Fairly new to X-Wing, and was wondering if you equipped a ship with both cards if you would get 2 free barrel roles during a turn. Since the Expert Handling gives you the free action, and then by completing an action, you can use EI to perform another action from an equipped card.

Or does that fall into the one type of action per turn rule?

The once-per-turn rule prevents it in this case.

As mentioned you run into that once-per-turn rule.

Something to note is that Expert Handling can trigger Experimental Interface but EI can ONLY be used to trigger an upgrade card that uses the Action header in it such as Expert Handling. When you use EI + EH you are almost always going to perform some action other than EH first; that then triggered EI which then allows you to use EH. If you use EH first you still trigger EI but are unlikely to have another upgrade card that requires an Action to use.

There are two somewhat popular upgrades that can allow multiple actions: EI and PtL (Push the Limits). Both can trigger anytime the ship performs an actions. Where they differ is in the type of action allowed. PtL ONLY allows a free action from the ship's action bar although there are some upgrades that do add actions to that. EI ONLY allows an Action from an upgrade card itself.