Temporary mounts can be bought from Zeke, they last a day or 7 days. The Crown Shop sells permanent mounts for both gold and crowns. Red and Blue Dragons, Ponies, the big Cats and some brooms can be bought with gold.
Some bosses drop temporary mounts (almost always 24 hour). I don't think any boss drops a permanent mount. I have heard people say they got a 24 hour mount from a silver chest, but it never happened to me.
Keep in mind that the bigger, more spectacular mounts are clumsy to maneuver in tight areas, and can get you drawn into unwanted fights. Brooms, wings, and the cat mounts are generally the most agile. If you are not sure how much difference it makes, try experimenting with 24 hour mounts on marleybone rooftops or in sunken city.