While a realistic scenario would be the WEAV designs referenced by a4android, the OP is clearly hoping for some sort of aerodynamic vehicles.
The problem is actually multifold. The atmosphere has no free oxygen to allow the use of conventional jet engines, the distances to cover are orders of magnitude longer than that of Earth (imagine flying from an air base in Minnesota to a spot over the Indian Ocean before you even encounter the first enemy craft, then having to dogfight, then flying back. Without air to air refuelling your "fighter" would be the size of an A-380
A-380 in flight
Finally, the atmosphere of Saturn itself is very violent, with some of the fastest winds recorded in the Solar System. Planes caught in a downdraft might have a real possibility of being sucked into a layer of the atmosphere where the pressure exceeds the aircraft's "crush depth", so instead of air war we are looking at a weird hybrid of air and submarine warfare.....
However, there may be some ways to handwave this. If you can get to Saturn in the first place, there is probably some compact energy source like fusion, so a suitably large airframe coupled to a suitably small fusion reactor would make a sort of rocket or ramjet powered craft (the fusion reactor can be heating atmospheric gases, for example). This is similar to the proposed nuclear powered aircraft designs of the 1950's.
Nuclear jet engine
I would expect the aircraft to be at least the size of an F-15 "Strike Eagle" to house the fusion plant and any associated equipment.
Because you are working in a very complex fluid dynamic environment, the airframe would need to be able to "morph" appropriately. If you are in the lower, denser layers of the atmosphere, you need less lifting surface, while in the upper layers, you will need far more lifting surfaces. As well, you will need a strong shape to resist "crush depths" should you be forced down for whatever reason. A large "blended wing" lifting body design is probably a good starting point:
Boeing X-48 Blended wing design
Because of the added complications of atmospheric pressure and turbulence, regular weapons like air to air missiles or SFnal weapons like lasers would have limited utility (they would not have sufficient range, for example). This might be worked around by arming the craft with a powerful rail or coilgun. The projectile could have limited ability to be guided onto target, but as the range increases the effects of atmospheric drag and turbulence would eventually break the target lock and the round would fall into the depths of Saturn. Explosive warheads would not really be necessary.
One issue which you have not brought up is where these craft are going to be based, or why they need to fly and fight in the atmosphere anyway? The design I hand waved could serve as an SSTO and fly to and from space to carriers orbiting Saturn, but the "why" is pretty open.