you're getting a version conflict. "Application not installed--an existing package by the same name with a conflicting signature is already installed" is produced normally when you try using another app store (1Mobile Market, Amazon Apps, SlideME) to update an app you downloaded from yet another app store (Google Play Store). this has nothing to do with Google Play Services--which is required for Play Store and other apps (Play Music, Google+, etc) to connect properly.
I am guessing that you tried sideloading the Play Services apk, downloaded online somewhere, and it attempted to replace the existing app which was likely preinstalled on your phone already, hence the conflict. it appears you already have Play Services but this will not include Play Store. many cheap devices have Play Services but lack Play Store.