Absolutely you need to relegate your current camera to snapshots. If you're looking to get into it on a budget then a used Nikon D7200 is a very good sport/bird camera body. It ha reasonable speed and a above average buffer (about 100 jpegs). You can pick them up used for about 500$ I just did a quick search on one of the forums and they had a 2 year old D7200 and a Nikon 18-140mm lens for $750, just to give you an idea of prices.
In the Nikon line (sorry, dont know about Canon) the next camera up and the D7200's replacement is the D7500...about $900 new for the body only and then of course the D500 which is just under $1500 for body only. All are DX cameras.
If it were me, and it was me a few years ago, I would try to find a D7200 if I was on a budget (I actually found a D7100 about 4 or 5 years ago for the same reason, paid 350 or something like that, but it is not as good as the D7200 buffer size)
A great and inexpensive lens is the Nikon is the
Tamron AF 70-300mm f/4.0-5.6 SP Di VC USD XLD. It's light, fast focus, stabilized sharp and very good at 300mm. It might be my favorite all around tele lens and I have about $15k worth of lens to choose from.