You can do this by making Gear Sets.

From your character equipment menu (default hotkey: "C"), there is a button for your gear sets right above your character portrait. This will bring up a new menu for your gear sets. At the bottom there is a plus-sign button that will create a new gear-set; this set will contain all of the equipment you are currently wearing.

Now just switch to another class, with your best equipment and make a new set. You can now double click on either set to easily switch between them. I believe you can also drag them down to your Hot-bars to easily switch to that equipment set with just one click!

If you upgrade the gear in your set, you can update it either with the update button on the character menu, or by right clicking on the set's name and choosing to update the set with your current gear.

Hope this helps!