What a website can see
A website can normally see the source IP used to reach it. That address may identify an ISP, ASN and approximate area. The browser can also expose language, screen size, software information and other values. Try My Info for a live demonstration.
VPNs and proxies
A VPN routes traffic through an encrypted tunnel to an exit server. Websites usually see the VPN server's IP. A proxy can also relay selected traffic, but its security and coverage depend on the protocol and application.
Router and firewall
Keep router firmware updated, disable unnecessary remote administration and use a strong unique administrator password. NAT is not a replacement for a firewall, and IPv6 networks should use appropriate inbound filtering as well.
Your IP is only one identifier
Signing in to an account, accepting persistent cookies or using a distinctive browser setup can identify or correlate activity even when the IP changes. Use unique passwords, multi-factor authentication and sensible browser privacy settings.
HTTPS still matters
HTTPS encrypts application traffic between your browser and a compatible website. A VPN protects the path to the VPN provider, but it does not replace HTTPS, software updates or endpoint security.