How to get latest version of Google Play Store I'm trying to solve a problem I'm having with Google Play Store. (Specifically, a continual "Error checking for updates" and an empty "Installed" list. I tried clearing the cache.)

The version I have of Play Store is 14.2.58, and I know it's not the latest. I've read that sometimes the app doesn't update automatically. But where do you go to upgrade the app manually?  

When I google for it, I see a lot of third-party sites that offer the latest download, but I don't know if they can be trusted. If Google itself offers the download, I can't find it.
If you don't trust other sites, then use Yalp.

Yalp Store lets you download apps directly from Google Play Store as apk files. It can search for updates of installed apps and lets you search for other apps. Yalp saves downloaded apks to your default download folder. Other features include browsing categories, viewing and leaving reviews, black/whitelisting apps for updates, filtering apps by being free/paid and containing/not containing ads.
To supplement the features related to Google Play, Yalp Store has ordinary package manager features: listing, running, installing and uninstalling local apps

Also, visit What are the alternative Android app markets? to get a list of other apps/places/sites that can help you get it.