I think that the easiest type of increase in Tunisian crochet is the yarn over increase. You simply make a yarn over in between 2 stitches and you consider this yarn over as a new stitch. I used the Tunisian simple stitch for the sample here. But it works with any type of stitch.
To make a yarn over, wrap the working yarn around your hook, from back to front, over the hook. At the start, the working yarn is at the back of the hook. After the yarn over increase, the working yarn is back at its initial place.
You can make this yarn over increase anywhere you want in a row. In the photo below, I first made a yarn over increase systematically before each yarn over increase from the previous row. Then I made a yarn over increase always after each yarn over increase from the previous row.
