Hi Christine,
You also say you have a 20D, but that you prefer the 300D for birds. I have a 300D, but not a 20D and my chief reason for wanting a 20D is to cope with birds - especially in flight.
Yes, you probably need to use burst mode. The 20D will do this a "million" times better than the 300D. Indeed, if you use RAW, the 300D will virtually not work in burst.
AI Servo is available on the 20D.
The information given on the A1 applies to the 300D - "P" is fully automatic. I, like you, tend to use Av - aperture priority and choose an aperture to give appropriate depth of field. While that might allow you to maximise depth of field by shutting down the aperture, that might lead to blur from movement except in very good light. Try Tv - shutter priority - and select a high shutter speed.
Focal length - can you get in a closer position where you can use a shorter focal length ? This is always a huge advantage, not just because the bigger magnification exagerates all movements, but also because you reduce the chances of bad air distorting the image. On a warm day this can be very noticeable even at short range over some surfaces, like stones. I guess you won't be shooting over tarmac, but think in those terms and you will know what I mean.
Focussing - you have two possible focussing strategies :
1. set your lens to manual focus, focus in advance on where you expect the bird to be and try to release the shutter as it passes thriugh that point. Using the burst mode gives another way of achieving this. However, I know a guy who gets stunning results using single shot, but he has a defined flight path to a feeder and keeps on trying and I think his rejection rate is 90% plus.
2. Use autofocus - which really means the 20D because the 300D is far too slow to react.
Image stabilisation - have you got IS switched off ? If not, that is what the book says for moving subjects, so I'd guess it's worth a try.
If you end up with under-exposed shots from pushing the shutter speed up to get rid of movement, and shutting down the aperture to get improved depth of field, don't despair ! Try auto levels etc in Photoshop.
Above all, keep trying, never give up. Study your failures - can you tell if lack of sharpness is due to focus (even blur), subject movement (smear) or shake (look for smear that runs counter to the subject's movement) ? If it's focus, think about Av again, if it's movement, you want a higher shutter speed. Noise is pehaps due to too high ISO - you may just have to tolerate that at the shooting stage. "Everybody" seems to think Canons are pretty good in this respect anyway.
Good luck
Mike.