You need to use a mixer to cut off the trail to your wallet. There are probably a few other ways, but that's the most straightforward one at the moment.
It should be safe enough. Running a VPN could add to the anonymity factor though, and might help with snooping Tor exit nodes depending on how you pull it off. Nord does say they don't keep logs, they're in an internet privacy friendly country, and that you can pay them anonymously with Bitcoin, so I'd say they could probably be trusted.
Have you already set it up? You should be able to pair it with a browser plugin. Check their website. You spend coins pretty much the same way you do with any other wallet afterwards.