You can also directly open a new shell session spawned by ssh-agent :

$ ssh-agent bash

The SSH_AUTH_SOCK and SSH_AGENT_PID variables will already be set in the new shell session. It will spare you exporting them manually (step one and two above).

I got my id_rsa identity added, still my verification problem repeats when I try to initialise push on GitHub. I use inteliJ>VCS>Import into Version Control>Share Project on GitHub. This is the error I receive:

Successfully created project 'ProjektZ_keprim' on GitHub, but initial push failed: Warning: Permanently added the RSA host key for IP address '140.82.118.4' to the list of known hosts. [email protected]: Permission denied (publickey). Could not read from remote repository. Please make sure you have the correct access rights and the repository exists.

And more, after following steps advised above my Git Bash throws:
ZuzannaRo@ZuzannaRo24 MINGW64 /
$ ssh-add -l
2048 SHA256:pbPkCPH/XRp5RPhI3dERAWR/OJN3HAHUMXilUByp1U4 /c/Users/ZuzannaRo/.ssh/id_rsa (RSA)

Save yourself the copy and paste job with eval.

eval $(ssh-agent)

https://unix.stackexchange.com/questions/351725/why-eval-the-output-of-ssh-agent

Save yourself another step and just do:

eval $(ssh-agent) && ssh-add .ssh/id_rsa

Also note that if you have cleverly used something other than .ssh/id_rsa for your SSH keys you'll need to change the command accordingly. (I use the filename <hostname>_rsa in order to avoid mixing up SSH keys from different computers.) You could probably do this automatically with cat ~/.ssh/config, grep, and some pipes, but that's way beyond my abilities lol.

Adding the following to ~/.ssh/config will help wit this. It also solves @elsiehupp's issue - i.e. there's no need to write code to do this automagically. Enjoy!

Host *
   AddKeysToAgent yes

Here's the manual page:

AddKeysToAgent
   Specifies whether keys should be automatically added to a running
   ssh-agent(1). If this option is set to yes and a key is
   loaded from a file, the key and its passphrase are added to the
   agent with the default lifetime, as if by ssh-add(1). If this
   option is set to ask, ssh will require confirmation using the
   SSH_ASKPASS program before adding a key (see ssh-add(1) for
   details). If this option is set to confirm, each use of the
   key must be confirmed, as if the -c option was specified to
   ssh-add(1). If this option is set to no, no keys are added
   to the agent. The argument must be yes, confirm,
   ask, or no. The default is no.

As a bonus, on a Mac you have the UseKeychain option (check it out in the man page on your Mac).