For a gold ranking in a bonus mission, you must complete all of the tasks within with a gold status. The specific objectives are found on an allie shown with a green streak as they fly around and are shown on your radar. Fly to them and press when prompted to start a mission. It will show you what rank you have and what's required at that time. There are a total of 4 different types of objective missions. They are all pretty easy and should take you only a few attempts on any of them.
Survival: You must fight off 5 waves of enemies. The time only counts down when the enemies appear on screen, and not while you wait for them inbetween. Use combos to get your flaming talons and this should be pretty easy.
Owlet Rescue: These are the easiest of all the missions. Just remember to use corckscrew which is up on when you are on the return trip since you'll be flying the opposite direction and need that boost. Any enemies that attack you are easily avoided with a barrel role.
Speed Flying: This is the most stressful of the events. You must fly through rings as you race to the finish. If you miss 3 rings you will have to try the whole thing over. Use corckscrew if you are flying slow or want a boost. You will gain lots of it while diving.
Bombing: You must destroy all of the turrets within the area. Avoid all enemies to make short work of these missions. The game gives you plenty of time.