How do you properly use more than 3 colors in a crochet project?
You use more than three colours the same way you use three colours.

If the different patches of colour are close together you can leave the thread idle while working till you need it again.
When slightly wider apart you can lock the idle yarn under a stitch or wind it around a different thread that is handy.

But if the distance is big, tie off and start with the colour again when you need it.

Depending on the actual style of crochetting yarn taken along under the stitches might show or be hidden.
If hidden, it is the best way to move the idle yarn for a few stitches, not only locking the lengths away but also strengthening the colour changes.
You can also use this method to anchor the end where you take out a colour and the start of use of a new colour.