VPNs will never be fully anonymous. However, there are things you can watch out for when choosing a VPN. First of all, check to see if the VPN your looking into operates within one of the '14 eyes' countries. These are countries that share intelligence with one another. So, for example, if your VPN is a Norwegian company, the US could ask Norway for your information, and the VPN service would give it up. However, they can only give up as much as you give them. There are VPN services that require minimal or no identifying information upon sign up. The best, in my opinion, is Mullvad. I would look into it.

Also, a red flag with VPNs is if they say that you'll be completely anonymous online. That's impossible. If they are willing to twist the truth to make money, what else are they lying about? A good privacy service in general will be up front and honest about their shortcomings, so you can actually make an informed choice.