Deleting all internet cookies/history/passwords for the steam client browser, my local PC's browser, uninstalling/re-installing steam, checking port forwarding on my router, etc. If the problem persists, you need to try logging out and back in again. You can immediately try to launch and sign in to the Steam client after turning off the firewall.

Turn off the firewall by clicking the button at the bottom of the screen. One of the most common and irritating errors while opening the Steam client is the 'verifying login information error. I created a dummy account and was able to authenticate without any issues. Allow changes to the settings by entering your admin login and password.

I verified it was an account issue as I tried logging in even from my cell phone's browser without being connected to wifi so I could be sure I was using the cell network to rule out local network or client issues. I could not submit a support ticket due to this problem as it would again try to do the login verification and say it could not reach a login server. I'd authenticate and put in my steam guard code but it would just go back to the login page (steam client would authenticate and show me logged in). In fact, in my case I could not even log in to steam via a browser. I was able to log in to steam from the client but my wishlist was inaccessible, I could not buy games due to the login verification popping up and failing, etc. In case anyone else is searching the web and finding all the solutions provided didn't work for them, I had a completely separate resolution: I had to remove steam guard and re-add it. Connection problems involving Steam or one of its games are usually fixed by doing a simple trick restarting the Steam client.

Steam wont verify login information