Q: How to log out and back in as admin?

System Preferences -> Users & Groups

Unlock the padlock in the lower left corner

Click "Login Options"

Change to automatic login to "Off", unless you still want that on, then pick the user to automatically login to.

Change the display login window to either a list of users you select from on the login window, or require they know and enter their username.  In both cases with automatic logins off, they will need to enter the correct password.  The username entered can be the long Mac name or the short Unix name for the user.

You do NOT need to logoff before shutting down.  You will be logged off automatically during shutdown.  The reboot login window will be whatever you specify in the Login Options above.

If you wish to become the Admin when you are logged in as a Standard user, you can use "Fast User Switching" and keep the current user logged in and switch back and forth using the Fast User switching menu bar item (if you do not have much RAM on your system, this may be slow and painful, but at least try it to see what you prefer).

Or you can logout of the standard user, and you will get the login screen, where you can select the admin account.  No need to shutdown in between switching users.

Logging out or Fast User switching works whether you have Automatic login enable, or you make the user enter their username and password at the boot time login screen.

1. If I were to simply log out of my admin account when shutting down, would I then log in by simply inserting my full name and admin password upon startup?

Yes, but you do not need to actually logout when shutting down.  You will automatically be logged out when you shutdown.  It is best to save any work you have in open apps before shutting down or logging out.

2. I have also read that setting up a Standard User account and using it would perhaps be a better way of handling this "situation".  Is that indeed the case, and

    what's the advantage of that procedure over simply using my admin account to log out and back in with my admin password?

Yes in the absolute sense, but for the most part if you are careful about what you do while logged in as admin it is not all that dangerous

A Standard user does not have any admin rights.  So you will need to switch to the Admin account or when prompted enter the admin account name and password before you can do things like install software into privileged locations.

Mostly this is just an extra step, and it is most useful when it is really another user (child, spouse, etc... that is sharing the Mac).  When you wear both hats, you can still do stupid things, by saying "Why Yes, I do want to install that Adware package, it sounds like such a good idea", and you proceed to switch to your Admin account.  Just a little more work, but since you are in reality the admin, you can switch to the privileged account.

boot with the Command key

Try the option/alt key during a restart.

Press the alt (a.k.a option) key at boot to select a startup disk.

You can also boot to recovery mode (cmd+r at startup) to select the startup disk, there are many ways to get out of many messes you can imagine .

Standard users are a little safer if you use auto login. An admin account retains admin privileges when you login for 5 minutes IIRC. That means they can tweak system settings during that grace period.

If you want real protection of your data use FileVault to encrypt the disk, otherwise rebooting in target disk mode exposes the data (file permissions are a weak form of protection).

It's probably simplest for you to create a new admin account & then use that to remove the admin rights on your current account. It should be surprisingly trivial to do