Most mid range point and shoot cameras will do 1080p at 60FPS, but if I were you I'd get a lower end dSLR. It will provide superior image quality and be around the $450-$600 range. I'm partial to Sony dSLRs, but Cannon and Nikon are the more popular ones that will have tons more options for additional gear. You could also go the GoPro route and spend a lot less than a dSLR, but lose some of the features. Many a YouTubers started out with a GoPro or a cheap point and shoot camera. Many mid range tubers use dSLRs.
Hope that helps in some way.