I can't get my 2005 Mitsubishi Triton into either two- or four-wheel drive

There are a lot of possibilities here. Either there’s a problem with the actual shifter for the transfer-case (that selects two or four-wheel-drive) or there might be something amiss inside the transfer-case itself. But before you try anything else, make sure that the transmission is in Neutral. Many vehicles will not shift into or out of four-wheel-drive while the transmission is in Drive or Reverse. So stick it in Neutral and see what happens.

From memory, the 4WD selector in your ute is a mechanical lever, not one of those newer rotary knobs. So at least you won’t be trying to overcome an electronic selection problem that occurs in these later versions.