I started a new game and am now at level 20 and I have not encountered any dragons yet. Is this normal? I don't recall having to search them out with my previous character, they usually just showed up randomly.

Is there a quest I need to start to start the dragon born stuff?

have you retrieved the dragonstone for the court wizard?

Yeah do the quest and then dragons will appear everywhere.

Yup, they won't appear until you do "Dragon Rising" from the main quest.

Sometimes they dont spawn until you go with Delphine to Kynesgrove.

hmm depends if you fast travel alot, i dont unless its city to city on a carriage and by using the fast travel method i've encountered about 3 in (one every other travel roughly)

from walking around, exploring and stuff i've encountered one and im now level 30.

This. On my 2nd playthrough I only got 1 random attack until I did A Blade in the Dark. Even wrote it down on paper.

This^ assuming you have activated the quest that starts random dragon spawns, you can still totally avoid them by fast traveling as soon as you exit a town or dungeon. (of course not counting the rare town attack, or even rarer dragon spawn in your face outside a cave, but for a great majority of cases.)