Does this tablet have google playstore and can i get games on it

normally updating apps are in the android market app or google playstore app (if an app requests for google play services to be updated tap the notification and it will bring you to update the playstore services)

it appears the google services app may need resetting (android settings > app manager > google services > clear data) this will reset the tablet's memory of whats installed until the playstore app is opened again if it fails let me know

In order for your tablet to have a working Playstore, it must be compatible with Playstore.
The reason that I? telling you this is because a lot of people buy a used tablet, with third party software in which force a Google app into the tablet.
Since there is no detail about the tablet, I can only guess.
Go to settings>app. From here search for Google Services, if there is none, then it means that you must install it for your device or update it. If there is none for your device, then it means that your device does not support Playstore.

Hi there joe i want you first of all to check your internet connection if it is connected please check the storage or if the playstore is restricted in your country if not try uninstalling then re installing or just try the app if you didn't install it yet or do contact thier support center i hope this answer was helpful

download games

Go to Google playstore. You should find a load of free games there.

I suggest that you troubleshoot first, instead of a hard reset. Factory reset is within settings.
If any google app won't download nor install, it's probably because your google services are not updated or is corrupted. Try the next steps before factory reset.
1. Go to settings, scroll down to app.
2. From here search for PlayStore and tap it.
3. Tap Clear Cache and Clear data.
4. Go Back.
5. Search now for Google services, this one is not under the download tab, which you can notice at the top of the screen, instead got to All and scroll down until you find it.
6. Tap Clear Cache and Clear data if available.
7. Reboot phone or tablet.
Most of the time this procedure do work, some people also opt to remove the google account and sign in after the reboot.
After the boot, open the PlayStore, accept the agreement and try to download anything from the PlayStore that is compatible with your tablet. Also, just to be sure, go to settings, security, look for Unknown Sources and enable it. This feature allow you to install any apk file that is not from the PlayStore.

Your tablet is more likely a chinese no name one.
You can download the Play Store standalone version from here:
APK 12 FEB 2015 Google Play Store v5 2 13 Newest version unmodded

Make sure you have a wifi connection. Playstore is on the internet so you have to be able to access the internet.
Also, if the Playstore app that came installed on the tablet is lost or needs re-installed for some reason, then you won't be able to re-install it on your tablet unless you also have gmail running on the tablet or google+ or some other app that Google can use to identify you as the tablet owner.

If they are using a mobile device management system like Amtel (the best one) then you can't, at least not without your school knowing. If your school went for a poorer MDM solution, they should have got Amtel - because then it could turn off games when you're at school, and turn them on again when you go home.