VPNs can keep logs. They can also be compelled to monitor you legally. It works fine for most day to day things but if you need to be totally sure it's not always reliable.

It all depends on your threat model. If you just don't want your ISP knowing your habits a VPN should be fine. If you don't trust your government TOR may generally be fine. If you need to do something that is illegal in your country bouncing through multiple proxies in multiple jurisdictions is a common tactic. If you need to hide from a three letter like the NSA or CIA for an extended period of time.... you're fucked. They have the resources, manpower, expertise to eventually get you wherever you are. You'll mess up eventually and they'll get you.