I think the fastest/easiest route is probably buying a perm Enchanted Broom from the crown shop for 50k gold. Alternatively there's fishing, which maybe isn't super easy, but you can get some nice/rare mounts if you have the patience and fish on castle tours (e.g. Botanical Gardens has the Celestial Wolf mount).
You can buy a few different types of basic mounts from the Crown Shop for gold.
You can craft mounts from Aquila (60,000 gold for the recipe, plus you must do some fishing and gardening, and Daily Assignments faithfully until you get a Mote of Transport reagent). You can also craft two mounts from the Boxes events (the red phone box dungeons that show up from time to time).
Mounts drop from many of the special boss battles and skeleton key battles. For example, I got the Mander Palanquin from Ra, a low level, wooden skeleton key boss in the library of Krokotopia.
Hi, i noticed that there is actually a few mounts that you CAN fish for at certain houses. Just go to the castle tours place and look for the house. Some mounts you can get are:
Aquilan Dual Chariot from the Acropolis Battle Narwhal from the Polarian Shipwreck Arcane Minecart from the Midday Estate Bee Wings from the Amber Estate Feathered Raptor from the Serpentine Escape Magic Carpet from the Nomad's Camp and finally Tyrannosaurus Rex from the Pyramid of the Lost Horizon
They are EXTREMELY rare, so don't get frustrated if you don't catch any within a day. Believe me, it takes ages.
Apart from the various mounts you can buy from the crown shop for gold. Many of the Skeleton key bosses drop mounts as a possible reward. If you are still low level the wooden key boss Ra in Krokotopia is likely best for you, with 7 perm mounts available.