Just to walk through this again to make sure you have done it properly.

1. Make sure you have a Google account.
2. Log into the developer console at: https://console.developers.google.com/
3. Create a project
4. THEN, select YouTube data API
5. Click create credentials on the right hand side of the screen.
6. Choose where you will call the API from
7. Pick public or User.
8. Obtain your API Key.

Optional: restrict your key.

I just walked though all of this and have been given a key myself.

Make sure that the YouTube Data API is enabled in the Google Developer Console. It should look something like this:

The key you are looking for is the server key, which looks something like this:

(I've truncated my key for security purposes). I now copy and paste this into my browser window:

https://www.googleapis.com/youtube/v3/search?part=snippet&q=YouTube+Data+API&type=video&key=YOURKEYWOULDGOHERE

This works for me. Start here to verify that the key is working and that the API has been enabled correctly. I suggest using the API client, because some API calls will require an OAuth 2.0 access token, not the simple API key (search API requires regular simple API key).