I use the ElGato Game Capture HD60 Pro for speedy videos, but the streaming side is not as good (from what I hear) and I don't stream. I can go into a lot more detail about my setup if you need to know what works really well.
For higher quality videos I use a BlackMagic DeckLink 4k Extreme 12G (~$1400), which is capable of up to 2160p60. I use a lot more hardware when I make 4k videos, and my Internet connection does not support uploads at that resolution so you're not likely to see that on YouTube anytime soon. My connection does not support streaming, but if I was going to do that I would probably use the 12G and canned videos (advertisements and detail videos) to switch, mix, and render, because with enough hardware it can really reduce your work load and the time involved in making videos. And here again I could write a lot more about this setup, but unless you have thousands (literally) for this it's not going to appeal to you.
EDIT: If I had it all to do over again I might just stick with AJA and get their KONA 4. As it was I had just bought a new M3 and didn't want to lay out the coin for the KONA when I bought the BlackMagic, but the price has dropped somewhat since.
The ElGato is all the casual user needs.