Cannot login after upgrading from 16.04 to 18.04

I recently tried to upgrade my laptop from 16.04 LTS to 18.04 LTS. The upgradation process was smooth; but when I try to log in, it is showing an Authentication Error

and then I get this black screen which stays indefinitely

I am a rookie in these regards, so a step-by-step guide will be much helpful. Thanks in advance.

Results after trying suggestions by Manu Mathur:

sudo apt-get update gives this:

sudo apt-get upgrade gives this:

Solution 1:

Had similar problem upgrading from 14.04 to 16.04. Logged to TTY and did a shutdown, which lead to other problem. Could log in to TTY anyway (CTRL+ALT+F1). Then I ran:

sudo dpkg --configure -a
sudo apt-get update
sudo apt-get upgrade
sudo reboot

Helped in my case. Solution is based on a related earlier answer. Configuring did the trick.