You have to figure out what you are trying to hide. Cookies will be cleared with Chrome/Edge in incognito/private mode.
The VPN connection protects your connection history as that is stored on your router and the gateways you pass through on the internet. With a VPN, the gateways, including your internet or cell provider, will only have a log of the connection to your VPN access point. Also make sure that you pick a VPN that doesn't store logs (Express, Nord, PIA, etc. - just google "no log vpn")
I recommend using a common separate browser like Firefox, when you want to mask your activities. It's not "the best" it is just easy, and you need to be consistent. You also want to hide in the crowd and not draw the question of "Why is he using TOR?" which is a marker for illicit activity. (TOR is better albeit much slower).
You can also avoid the Google and Microsoft search empire by using search sites like DuckDuckGo to avoid non-cookie-tracking (ID targeting).
If you are doing hard core evil, then please depart this site, and activate your Nord/TOR connection paid for with Bitcoins so we can remain safe without you.