Hi. Depends what you mean by "not too pricey"
I was in your position about 18 months ago. I got a Canon DSLR kit as a present which was based around the entry-level 100D body but the kit lenses were relatively poor in terms of reach and quality (the biggest was a 75-300mm). It quickly became apparent that a much better lens was required.
Although there are camera bodies which are much higher-specced than mine (I'll get one... one day!) it seems that the current entry-level Canons (sorry, can't speak for Nikons) are specced at about the same level as mid-range Canons of around 4-5 years ago.
I think the current price for the 100D body (probably with a basic 18-55mm "standard" lens) is about £250.
Most of your budget should be reserved for the lens. Buy the best you can afford. If you can stretch to it, the Canon 400mm f5.6 is a great lens for static and "in-flight" shots and for even more flexibility (and with image stabilisation) there is the Canon 100-400mm mk2 zoom which I eventually plumped for. However, these lenses are, indeed, "pricey" (the 400mm f5.6 is about £850 - the zoom is about 2x that).
Because the 100-400mm mk2 is a recent model, there are lots of second-hand mk1 models becoming available. The second-hand market is also a great place to look if your budget is a bit tighter. Try
www.mbp.com
Sorry, I can't comment on "bridge cameras" myself but there are good alternatives there too.
Happy snapping!