A VPN
Your traffic goes encrypted to the VPN provider's server, which makes the request on your behalf. Sites see the VPN's address; your internet provider sees only that you are talking to the VPN.
This genuinely defeats address-based tracking by the sites you visit and monitoring by your provider or the network you are on. It does not make you anonymous: your VPN provider now occupies exactly the position your provider used to, and everything you log into still knows who you are. Choose one whose logging policy you have actually read.
Tor
Tor routes each connection through three volunteer relays, none of which knows both who you are and what you are asking for. It is a much stronger guarantee than a VPN, because it removes the single trusted party.
It is also slower, blocked by a fair number of sites, and easy to undermine — logging into an account over Tor identifies you to that service just as completely as it would otherwise.
A proxy
An HTTP or SOCKS proxy relays your requests without necessarily encrypting anything. It changes the address a site sees and does nothing else. Free public proxies are a bad trade: the operator sees your unencrypted traffic, and their motive for running one for free is worth thinking about.
What none of them fix
Your address is one identifier among many, and not the sharpest. The combination of your fonts, screen size, timezone, language settings and headers is close to unique and travels with you across every exit node you use — that is browser fingerprinting, and our connection page shows you how much of it your browser hands over.
Two specific leaks are worth checking after setting any of this up: WebRTC can reveal addresses your device holds directly, bypassing the tunnel, and a timezone that does not match the country your new address is registered in is a visible contradiction. Both are shown on the connection page.