How can I navigate my scene FPS style similar to how Unity does? I am aware of ShiftF fly, but it's extremely cumbersome having to use the scroll wheel to change the velocity of flight and moving the mouse from the center to spin.

Is there a way to move the viewport camera (not the scene camera) like Unity lets you right click and fly around with WASD and moving the mouse?
The latest version of Blender has Walk Mode, You can access it by pressing Shift~ (the tilde/backtick key above Tab) or View > Navigation > Walk Navigation. If you want to use Fly Mode (which used to be the default), you can switch between them in User Preferences > Input.

Under 2.8+ you can access both from the 3D View header menu View > Navigate > Walk or Fly:



It's WASD just like in FPS-games, Q and R to move up/down. G or Tab enable gravity and collision, and V to jump in gravity mode. Space teleports you to the surface you are pointed at. You can even change the key-bindings to enable FPS Walk (the official name) on MMB down, and confirm on MMB up to always rotate (or rather: look) in FPS-style - or even hold MMB down and use WASD!