I'm surprised I haven't seen this one more: Lizaaaaard You can purchase a combat trained one for 550gp from Animal Archives.
I believe, based on my reading of the purchase guidelines, you can buy this after your first game (most level 1's give about 550). Large mounts are annoying in some scenarios, sure, but even if it has to squeeze all the time it will be fine. Nothing like a CR 5, 7 HD monster as your mount starting with your second game.
Also the: Vulchaaaa is allowed. Though substantially weaker and more expensive, it let's you fly. At 1125gp you can still scrape together the money before level 2.
If you are going for pure output glass cannon cheese: Tiga! Tiga! For only 500gp you can have a moderate example that lovely pounce/rake kitty your druid friend loves so much and that he hast to wait until level 7 for. It's AC is terrible; it will die, but you bought it for the alpha.
When you are done messing around and would like your forever friend you pick up: BFF Hello +16/4d8+18 charge. Hello trample. Hello +15 perception/scent. Hello 79hp/diehard. Goodbye 3000gp.
Note, with all of the above, prepare to have your GM force handle animal checks and/or ride checks. Consider picking up a War Saddle (and saving a bit of money since you don't need to buy combat training). If you are not a melee type and thus have a free belt slot the Equestian belt will serve you even better and is cheaper.
If you are not planing to ride then the Collar of Obedience is likely your goto item instead.
Also an option, if you are using a wayfinder, is the Turquoise Sphere and slot it for +5 ride, +2 handle (stacks with Collar).