Cannot get Google services and play store to load in the apps area in the settings.

I have a ZTE Grand x 3 with Android version 5.1.1. Am unable to get services to work properly as it is not in the list of installed programs in the apps area of settings. I can find it under all apps and it is running. I have cleared the cache but did it while it was running. I can't get the play store open at all and go to the net to use play store. But Google chrome stops or gets hung up and stops responding so I tell it to wait and then I am able to move again. I was able to get the play store back in the apps installed but it gives me an error stating "google won't run unless you update Google play services.". I can't find Google play services to update. I have tried to use your APK and install the services but it fails every time. I truly don't want to do a factory reset as the information on the phone is immeasurable. I have been backing up to the SD card but can't verify that I have everything (all system and personal data) backed up. HELP!!! What is my next move?