Sbpr
Which sources should I be using? Just Github/Gitlab or should I be fetching from other sources too such as the developer's website or from an F-Droid repo
I'd say most of the sources in Obtainium. I would imagine that sources like APKPure / Aptoide / Uptodown might not be safe / reliable to use. But in general, the purpose of Obtainium is to be as source-agnostic as possible, so people have a convenient app installer to use. Being able to use multiple sources is the point.
I've seen at least one comment that made it seem like using Obtainium to pull from F-Droid repos was better than using an F-Droid client., but they didn't explain why.
I think you're referring to my comment? Being honest, not sure what in my comment "made it seem like using Obtainium to pull from Fdroid reos was better". That particular comment was just listing ways to install the app. With that said though, Obtainium is better than using Fdroid.
Obtainium allows you to install apps directly from Github APKs which (IMO) is better than installing from Fdroid. Not only that, but you can still use Obtainium to install apps from Fdroid repos (for apps that don't have Github APKs) so you don't have to have both Obtainium and [insert Fdroid client of choice]. Its a matter of convenience and using Obtainium as a central "all-in-one" hub for my (non-GPlay) apps.
which installation method and APK variant should I choose first?
As others have said here, either Google Play or Obtainium for installation method (eventually Accrescent).
For apps with multiple APK variants, almost always ARM64-v8a. If there isn't one, then either the APK with "aarch" in it or the "universal" one if they are available. If by "variant" you mean APKs with "Fdroid" / "Gplay" or "FOSS" / "Full" or something similar, then its up to you to look into the differences of the particular app and decide which version you want.
Are there any reasons for why it would be better to install apps from somewhere other than the Play Store in a profile that already has the Play Store installed while logged into a google account?
Certain apps aren't allowed on Google Play (e.g. Newpipe). Its also possible (albeit unlikely?) that, for whatever reason, apps installed on Google Play could be removed from the Store in the future. Imagine the Fossify apps potentially being dropped from GPlay. In this situation, you can still rely on their Github APKs (or Fdroid if you wish) to install / update the apps. Personally I only use Google Play for a few apps (less than 10 apps, one of which is my banking app) that I can only get from Google Play.