I want to be a dragon owner and raider but how to do it if I'm not Targaryen? Is there any way to do it, like with the Valerian sword quest line?
If you are in the Seven Kingdoms and are appointed to the small council, you can try to "face the dragon" if any reside there and are riderless. An event will pop up offering the chance to release the dragon, try and slay it (bad relations with dragonriding targs results if you succeed) and to tame it (depending on skills, you have a high chance if injury, failure, and death by dragonfire).
If the Targ dragons are dead, there is a chance one will spawn in Asshai. There is also a very small chance someone in the world will get a dragon egg, in which case (if you are that lucky, I mean with the number of characters in any one game, this almost is as rare as winning the lottery, it seems) you can gather dragon intel by visiting locations like Asshai, Old Volantis, the Smoking Sea, Field of Fire (Hammerhall province), Harrenhall, etc.
Another way is to marry into a targ with a dragon. Either you or your children will likely either get a dragon or an egg. When i played as Aegon the Conqueror, I married my dragon riding heir to Orys Baratheon's daughter (she was a bastard with Valyrian features by a random, lowborn courtier girl).
Then her dragon (I gave her one out of the kindness of my woefully undersized heart) had like five eggs. After she finished handing them out to my grandkids, she gave them to Orys, her brothers, sisters, and all of their children.
Before I knew it all the Baratheons had dragons! Only many hasty assassinations stopped Robert's Rebellion from happening much, much earlier (his son was ambitious and hated my because he was really Argella's bastards and I tried to expose him to my buddy Orys.)