You can't go wrong with Nikon or Canon. I am a Canon snob, so I know more about them. And I am a very serious amateur photographer who strives to create nice images in the camera with the best gear I can afford to do that with. I don't take happy snaps. So everyone's needs will be different, consider that too
Look at
how much you want to spend and go from there. Also, take into consideration your wish to hand hold or not. Personally, I have grown to love hand holding my series 1 EF 500mm f/4L IS USM with a 1D camera body and I'll happily shoot all day while hand holding. I can also do that with the series 1 EF 600mm f/4L IS USM, though my shoulders ache a bit after for a few days.
My partner on the other hand uses a 7DMkII and the EF 100-400mm f/4.5-5.6L IS II USM with an EF 1.4x MkIII converter (and without converter) and is loving it, but still finds it a little heavy. But she is building up strength.
With Nikon, you could consider their newer 200-500mm f/5.6 VR lens with whatever camera body (I know very little about Nikon, sorry) as a good entry level lens.
My best advice is:
always opt for a better lens over a better body. Camera bodies come and go (
man, I never buy a new camera, all mine are bought off E-bay or friends and I have been using very affordable 1D series bodies for the past 5 years). Obviously higher end cameras will have better AF systems and better low light performance and handle digital noise better. There is a lot to photographing with DSLR cameras and I could write a book on my personal experiences and tips for those new to the "game" - especially about things like minimising noise, how to expose a digital file etc.
Lenses, however, last far longer and they make the image.
Maybe get to know other birders who also photograph. Go out with them or talk to them, try their gear if they allow. I would not spend too much time talking to camera shop sales people, as many are probably not bird photographers and would not have half a clue about what you need. I'd leave discussions with camera shops at bargaining to get a better price for what you want.
Also consider, that the latest and greatest of cameras and lenses do not make good photographers. Time in the field using one's gear and practising is what makes good photographers.
Enjoy the journey, it is an amazing experience!
Best wishes...