Hi,
It all depends what you want to do. I could recommend loads of things, but I'm not you.... so any recommendation would be ... less then perfect

Personally I started out much like you, point and shoot, then my dads camera, then I choose my first good SLR (yes ... film days and my first slr from Canon turned out to be a bad choice, bought a better one few years later) and then moved on to digital, the good old 300D (shot loads of shots with that). But I digress.
If I were you, you're not, I would go to a camera store and just have a look at the entry level kits. Canon 4000 with kit lens for instance (or the 1300) and a Tamron 70-300 VC di usd (has been mentioned before), it's a kit that won't break the bank and still has a pretty solid performance. It gets you into photography for airshows. The kit lens is a step up from your point and shoot (assuming it is a "normal point and shoot" and not a high level one). Any Canon lens there is, is either EF or EF-s (let's not go the earlier Canon route) there are load to choose from. But with a good kitlens (the 18-55 or 18-135) you go a long way. With that you got the static covered. I've choosen the Tamron 70-300 VC di usd for a reason, it is a solid performer. It is one I bought second hand and I still grab back to it cause it is such a nice handy all round lens and it has an image stabilizer. And will cover take-offs and landings easily. Want more reach? 100-400 (Tamron and Sigma being the cheapest here but way over your budget)
And don't forget a camera bag or backpack... often on the bottom of the list... but still, you got to protect your gear a litte
Offcourse it doesn't have to be Canon, Nikon is great too

So is Sony. I chose Canon a long time ago for 2 reasons, loads of lenses and I could use a convertor ring to use all my old M42 stuff (of which I still have Loads

) Just go out.... have a look, have a feel and see what you want. It's about you, you are going to shoot with it. And you still got time before the airshow seasons start. No rush, just look, compare and see what you need.