It should be understood that most commercial VPNs are not a perfect solution for anonymity. A VPN is a great form of protection, but who manages your VPN can be a problem. Here's why a VPN might not be for you.
This isn't to say that you should not use a public VPN in general. There are plenty of reasons for why you should use a VPN. These are only points for why a public VPN is not a perfect solution for anonymity. By "public VPN", I mean virtual private networks sold as a service.
It is a common misconception that VPNs encrypt data through all travel. Only the data being sent from your data to the VPN servers is encrypted. For the data to reach the intended target like a website server, the VPN needs to remove the encryption. Only by decrypting the data can the end server read the expected data. Websites won't magically understand the new encryption. Websites can only work with the encryption they were designed to use, such as their SSL certificates(HTTPS).
Commercial VPNs are just a glorified proxy. Although your internet service provider no longer has complete access to your traffic, your traffic is still at the mercy of another company. You're forwarding your internet requests to a VPN just so they can resend the request to the end target like a website.
No logging claims
Companies can get away with claims like this easily. "No logs" could mean they are only not logging specific bits that they don't need or even don't have access to. By saying no logs, it can mean they aren't logging your sent Facebook messages, but they can't anyway because they can't read them due to HTTPS encryption. It could be something less specific like not logging the exact website pages you visit, but simply log what website domains you visit. Many VPNs continue to log because there is a lot of money to be made by selling your data to advertisers.
Even if they push hard that they do not log anything at all, you have no way to verify this. This mindset may be a bit paranoid, but it is true. If you think all lying companies will go out of business for lying in this manner, you're wrong. One VPN company that heavily advertised itself as a privacy centered company turned out to share all sorts of data. More information can be found here.
Knowing that a VPN can decrypt the data they encrypted, using a VPN as a service offers very little in terms of privacy. You're giving your internet activity information to VPNs and they can do whatever they want and can with it. They have complete access to all internet traffic that wasn't already encrypted. Privacy is already gone if they're logging at all, which you never truly know if they are or not.
Limited internet speeds
A VPN can only respond with so much bandwidth. If you have an internet speed of 100 megabits per second internet(Mbps), a VPN may cap your speeds at 30 Mbps. Your speeds will almost always be noticeably slower. This can make your precious money spent on a high-speed internet plan wasted.
What can I do?
This article isn't saying you shouldn't use a VPN. This article is only saying you should be cautious of public/commercial VPNs. Instead, you should make a personal VPN using a cloud server. It can be much cheaper than most premium VPNs. A tutorial is coming, so subscribe for free if you're interested.
I desperately need a commercial VPN
If you seriously are not willing to or can't make a VPN, the only public VPN I can recommend as of August 2020 is ExpressVPN. Their key selling point seems to be their claims of very little data being collected. Though that is what the previously mentioned VPN did too. Either way, your data is still at their mercy.
Using a public VPN puts your internet traffic is in their hands. VPNs are nothing more than a proxy. If you hope for a higher degree of anonymity, consider a home made VPN. VPNs would hardly put up a fight to give your data to authorities. If you're only trying to mask your IP from internet trolls getting your location, then maybe you can use a VPN like ExpressVPN.