Unfortunately, it is not possible to logout of the GMail application on an Android phone or tablet. The phone/tablet is linked with gmail account and the first gmail account you use will be the primary account on the device and this can not be deleted.
The only option to change or delete a primary account is to factory reset the device and then use a different email id during the fresh setup(Note that you will loose all your data when you do this). Even if you manage to find a workaround to delete the primary account, the apps already installed may not work properly.
If you do not want your regular gmail account to be the primary account, the best thing is to create a new account(the phone lets you do this at the time of setup) and use this account for all app installations etc. In this scenario, you can add/remove your regular or other gmail accounts later because they are not the primary accounts now. But the problem here is the paid apps, you can only use your paid apps when you install those apps using the account you bought them with. For this reason I have a seperate gmail account for use just with my android devices.
If you do not want to see notifications, you can go to settings>>apps>>Gmail and uncheck "show notifications" If you do not want the mails to be synced, goto settings and select google under accounts. Then tap on the account you want and you will have all the different options for the account here. You can edit the options you want. If you do not want your emails to be seen by others, probably the best thing you can do is to install an 'app locking' app. There are plenty of such apps in the play store. Once installed, you can configure which apps you want to lock. When anyone tries to open the locked app, say GMail, it asks for the preset password to access(this is different from your GMail password).