Pretty much all flight on Earth relies - unsurprisingly - on gravity, so there aren't a lot of examples we can go with from Earth's flying species. We can get a few ideas from aquatic species that might be more useful, but air and water are quite different environments in terms of viscosity, so 'swimming' through air is unlikely to be useful. Still, there are probably analogs depending on the ecology of your world.
For this I'm going to assume that there are air currents of some sort in the gas envelope/atmosphere. This seems to be necessary to allow a proper respiration cycle and prevent atmospheric stagnation. I'll leave it to you to explain how those air currents work, I'm just interested in their effect on life forms.
Given those air currents we can start with microscopic airborne algae, similar to the phytoplankton you find in Earth's oceans, that forms the basis of your ecology. Drifts of algae will be carried along by the air currents, possibly forming visible dust clouds, probably limiting visual range in high density areas.
Along with the algae swarms you could add other lightweight plant forms, structured to be carried along on the breeze. They'd be fluffy, spherical masses with leaves (green if chlorophyll is dominant, depending on ambient light conditions) on the outside and root mats on the inside that capture particulates and moisture from the atmosphere. I imagine there's at least one variant of this plant that grows to be a large hollow sphere that floats along, possibly carrying small animals and other things inside and out.
Simple animal life can use the air currents as well, simply floating along with very little actual motion. Larger, denser bodies will be less affected by the currents, so in algae-rich areas the food will be passing by on the breeze. Passive capture of nutrients in this way could support fairly large animal forms, since their energy use is very low.
More active animal forms will need to develop from more mobile precursors. Amoeba with cilia and flagella for motion are fairly common on Earth, and there's no reason they couldn't develop in your world. Simple forms grow and become more interesting.
Now that we have the basics, we can start theorizing on the developments.
Floaters
Algae, plants and drifting collectors just need to float in the breeze. They don't need any motive power since their movements are controlled by the air currents. Some drifting collector-class creatures may have fins or other aerodynamic features to better orient themselves, but these can be purely structural with little or no control.
Floaters also includes baloon-type creatures, like the Portugese Man-O-War jellyfish. A lot of variation is available there, from large single bags to hundreds of tiny gas cells. Gas can be generated biologically, through symbiotic bacteria or drawn directly from the surrounding atmosphere using a bellows arrangement.
Sailers
Extend the aerodynamic surfaces on a floater and add more control musculature and you can better control movement through the air streams. Being able to tack against the wind makes a sailer more able to maneuver to greener pastures.
Undulants
Sails become active surfaces that use rippling motion to push against the air, allowing the creature to move against the airflow or in areas with low air movement. Think of how cuttlefish fins work, but much bigger. Also imagine ribbons and carpets that undulate in the air, using cilia on their surfaces to increase air resistance. Might look something like how snakes do when they're moving through water or over sand.
Jets
Simple balloons move well with the breeze, but with a little effort the internal gasses can be pressurized and used to maneuver. Over time a variety of animals of all sizes can develop from here, from simple mobile spheres to fast hunters that draw in air, pressurize it, then push it through directional nozzles. With the right sort of multi-stage pressurization this could even be a sustained flight model.
(Juvenile jet-based creatures would have to be strongly familial since they'd be completely helpless until they learn how to control their jets. Packs of baby jets would probably be terribly cute.)
Opposed Wings
While Earth-formed wings use gravity, wings in this environment would need to have some other opposed force. Rather than having pairs of wings pushing 'down' and providing aerodynamic lift, a set of opposed pairs of wings pushing against each other would work as well. This could work well for insect-like forms, but other large creatures could benefit from it as well. A bird with four wings and a pair of tail surfaces would work well.
Instead of simple pairs you could have a ring of wings where each wing opposes both of its neighbours during the full cycle. This would be quite maneuverable but potentially not as fast.
Flapping Umbrella
I don't know what else to call this. I looked, but I just can't find a good way to say it. Take an umbrella with open flaps along the surface. Open it up, letting air pass throught those flaps. Now close it, having the flaps seal against the surface. Net result: weird, slightly ugly propulsion along the shaft of the umbrella.
(I don't know, it just popped into my head. Probably thinking of octopuses. Octopodes. Octopi? You know, Sea Cthulhus.)
Those are all very basic concepts with plenty of variation and development possibility. Some creatures will undoubtedly use hybrid motion forms, combining a relaxed drifting option with high-energy movement capability when hunting or hunted.
