I've been doing airshows for over 30 years and I've used a fairly large variety of lenses during this span of time. My favorite lens, and the one I use most often now, is a Tamron 300mm f/2.8 LD-IF. You'll be hard pressed to find one for $300 though, so I'll discuss a couple of others that might work well for you.
First of all, I've been to airshows where 300mm was too much for some shots, and I've been to airshows where 300mm was not enough for some shots. But in most instances, 300mm is just about ideal. So this is the focal length I'd recommend. If you can find it in your budget to get something longer, then I'd recommend a good 500mm mirror, like the Tamron 55B or 55BB -- both are excellent and rival refractors in terms of sharpness and contrast. Pack this along with your 300mm, not instead of it.
The Nikon 300mm f/4.5 is a good choice. The 300mm f/4 ED is a better one, if you can find one for $300. Which might be tough. But there's a sleeper that not many folks are aware of that takes outstanding images -- the manual focus Tokina AT-X 100-300mm f/4 SD. It has the SD glass, so it handles chromatic aberrations well. It has f/4 speed at 300mm, which is great. And it is as sharp as some of the best 300mm telephotos I've tried -- even wide open! This zoom is what I would describe as "best in class" in terms of manual focus zooms that reach out to 300mm. And best of all, it can be obtained for much less tha $300. I have one in Nikon mount, and it has become one of my most favorite zooms. It's large, as you might expect, so it has a tripod mount. Now, for airshows, I usually don't recommend a tripod. A monopod is useful in some situations, but I usually forgo both.
At an airshow, when taking pix of a prop-driven airplane, you'll really want to be shooting at shutter speeds of 1/250 or slower. You get decent prop blur at 1/250, quite a bit better at 1/125. But the problem with shooting at these slower speeds is the potential for camera shake. So to prevent this, I've practiced smooth follow-throughs. I've found that a somewhat heavier optic actually can improve sharpness during panning because the weight acts as a damper against user-induced jitters that might be transferred to the camera. So don't be afraid of using a larger lens like a 300/4 or /4.5 -- or this Tokina -- because of its weight. Just be sure that your pans when shooting at slower speeds are as smooth as possible and you'll do fine.
