It's counterintuitive and misleading, but if you use YouTube's no cookies domain, YouTube will still set cookies when someone starts playing a video.

Try updating your browser or clear your browser's cache & cookies. You can also check whether your browser extensions that block ads are affecting video playback. As another option, try opening YouTube in an incognito window with all extensions disabled and check if the issue continues.

To search for videos, simply type what you're looking for in the search bar and click the Search button. Results will appear below, just like a Google search. If you want to get really detailed, click the Filter button below the search box.