The key to mastery of the SNES-era Fire Stingray ("the pink one") is the timing of the S-jet (Super Jet). You don't want to use that until you're absolutely sure you won't have to slow down for a while.... unless you really need it (exempli gratia, for your troublesome jump).
Piloting an F-Zero machine is not like driving a car. You have three digital dimensions of control, not one analogue one. Pull back on the stick [down on d-pad] as you hit the jump plate to gain height. Holding the stick back will keep you in the air longer, and keeping it held as you land will keep you from hitting the track (F-Zero machines, as you probably know, are designed to hover magnetically about two feet from the track) and help you maintain your speed.
The Fire Stingray is a rather heavy and bulky machine (weighing in at nearly 2,000 kilograms), and it takes even the twin RS5025 engines a little while to really get it going. The Stingray handles well around corners, so don't bother using the brake, just be sure to shift your weight in the direction of your turn and, if it's really sharp, punch the throttle a few times to fight the momentum.
