It only shows one of my Google accounts. I want to log into a different one but I don't want them connected in any way. What do I do?
If you mean multiple Google accounts at the same time...
In the top right corner of any Google window (mail.google.com for example), select your user icon and select "Add account". (next to Sign out)
Log into another gmail / apps account. Follow any instructions if multi-account login is not enabled.
Once this is completed, you can use that same process (select icon) to switch accounts.
The only 'connection' between accounts is that if you log out, you log out of all accounts.
You can be logged into 4 accounts at the same time.
If you don't want to use Kevan's suggestion, use a second browser profile.
For example, you can start firefox with the flag -p from a command line, and it will let you create a second profile. Then you can log in into the one account from one browser window and into the other from the other browser window. You will have to start firefox with the flag --no-remote to be able to open a second window with another profile. It is a good idea to use a different theme in both, so you don't mix them up. (They behave like completely separate installations, you must install all addons separately, and adjust all settings the way you want them).
Chrome supports profiles too. I don't know about other browsers.
Be aware that many sites you visit will gather lots of information about you. The fact that you are using the same IP address, same MAC address, and are logged in at the same time is enough for them to be absolutely sure that it is someone from the same household, and quite confident that it is the same person. So, while officially there won't be a known connection between the accounts, Google's algorithms which work with recognizing identities of visitors (e.g. algorithms for personalizing ads, for suggesting contacts on Google+, etc.) will know that there is a connection.