A few helpful mail tips

Most of these things never really bothered me on their own, but after some time it just starts to get annoying. Especially when all the mail you get is about how a file exists that you could easily and quickly delete, or how the configuration is just not right for another certain, convenient feature.

How do I read mail? / Which client should I use?

To read your mails, you could use the standard mail application. I suggest you try others, like mutt. It's actually quite nice, if, like me, you just need a client to read local system mail. When it comes to more, you might want to check if your favourite E-Mail desktop application can read the system mail.

Mailing to remote domains not supported

You should be careful with this: If you misconfigure the external access to your mail daemon, you could become a spamhost. Kris Leech posted something about this in 2010, he's also the source of this simple solution.
sudo dpkg-reconfigure exim4-config
Then, select the top entry: "internet site; mail is sent and received directly using SMTP". The next entry may be left at default, which should just be your system's name. In my case, I left it to be JuiceFridge8. The next one is important! "IP-addresses to listen on for incoming connections:" decides, who is able to actually send you mail. Leave this on the local addresses and ::1, that way it's limited to internal mails only. The next one may be left empty, it may also say something like "systemname.localdomain", which should be alright in most cases. The next one should be left empty, it decides which hosts to relay mails for; since we limited the system to receive mails only from localhost, this shouldn't matter. The rest can really be left on default values.

ALERT: exim paniclog /var/log/exim4/paniclog has non-zero size, mail system possibly broken

This actually has a very simple solution: Remove the paniclog file. It took me too long to figure this one out.. I didn't realize it said "has non-zero size", aka. "dude, the file is not supposed to be there".
sudo rm /var/log/exim4/paniclog
After doing this, check if the error still appears.
sudo service exim4 restart 
That's it! :)