Ah, okay. So the simplest case is that you just don't have those features through Google so you have to find another provider. For instance you can do contact sync through CardDAV courtesy of apps like https://www.davx5.com/, email with K-9 Mail, get apps from F-Droid, etc. Depending on exactly what the problem with Google is, you can also use unofficial software that still talks to Google services like https://microg.org/ instead of play services and newpipe instead of the official youtube app.

LineageOS comes with AOSP stock applications and you can install additional app stores on it. You can also install the google apps packages including play store and play services, but you don't have to. Of course many 3rd party apps rely on play services e.g. for payment.