1. You can get sigils which are the currency used to get a seasonal dragon from participating in events or from chest. Different seasons have different sigils. This season it looks like this:
2. You can check to see how many sigils you have by going to the season tab of an event and clicking on one of the blue box. The top number is what you have. The bottom is what you need to get the prize on the box.
3. As you go up the line you will eventually get a seasonal dragon. You need around 1500 sigils to get you first dragon.
4. After that you can continue on that line because it will lead you to evolution stone. They are used to evolve the season dragon at a certain levels. It looks like this:
There are multiple evolution stone. Choose a dragon, go alone the line and get the dragons and the stones to evolve it.