You can learn all about the different settings and change the aperture and shutter speed and ISO and shoot in RAW and get a huge, expensive lens...OR you can just keep it simple like I do. I recently switched from a bridge point-and-shoot to a DSLR, as you know, and I just discovered that the "sport" mode is great for birds. It is basically auto mode (for fast-moving subjects) and all you really need to know is how to set the exposure compensation and turn on continuous shooting. If you already know how to crop and lighten your photos afterwards (just to see the birds better), you'd be good to go.
My camera is the Nikon D3400, an entry-level DSLR. My lens is the AF-P NIKKOR 70-300mm f/4.5-6.3G ED VR. It doesn't have the most reach and it doesn't have a huge aperture, but it's really quick to focus and gets the job done. "VR" (which is Vibration Reduction in Nikons) is very important when shooting handheld (I'm sure your point-and-shoot had something similar). The photos I take with this combination are similar in quality to my Canon Powershot SX50's, but I can now take clear photos of birds at a moment's notice which is so helpful!
I bought the camera and lens off eBay. The lens was a little over $100 and the camera was probably around $200 (it came in a bundle), so it all cost about as much as a point-and-shoot.