For a simple introduction, I thought it would be better to skip the subject of tunings altogether and just focus on the structural implications that one would "read" in an actual piece of sheet music.
But in my opinion, how the notes that you play simultaneously sound together is the core of music theory. A good chord progression can feel like a journey and it will set the mood of the entire song. Therefore, I would argue that note frequencies and the reasons why certain tuning systems are preferred in certain situations are the key concept to understand if one wants to elicit a chosen mood in the listener.
But I brought up Just Intonation here because people tend to use that when they sing freely. So I would see this as a way to first understand why people (and famous composers) act the way they do before one tries to categorize things scientifically.
In the context of an electronic synth or keyboard, C# and Db are perfectly the same. In almost all traditional music, though, instruments are almost never tuned to equal temperament, so that C# and Db then mean different things.
As a famous example, Bach was very fond of his well-tempered clavier, which was a tuning method that is to be found somewhere between pythagorean tunings (usable for only one scale) and equal temperament (everything sounds equally lifeless). Bach's tuning was therefore a very carefully chosen compromise between being able to play the most common scales cleanly and sacrificing some scales in exchange for more precise harmonic relationships.
All famous European composers were experts at squeezing nice harmonics out of an un-equal tuning system where some key combinations just so happened to always sound horrible. It was called "Wolfsquinte" if you hit the wrong combination. If you limit yourself to only equal tuning, your are missing out on the slight harmonic differences that are the historical basic for all contemporary harmonic progressions.
If you start with one fixed frequency and then derive everything else as pure harmonics, that is called "Just Intonation". Wikipedia has a table on how those perfect harmonics differ from the notes that are mapped to your 12 keys: https://en.wikipedia.org/wiki/Equal_temperament#Comparison_w...
And lastly, by only thinking about notes as the keys on your keyboard, you completely lose the concept of musical Commata, which are when the true frequencies of two notes happen to fall onto the same key on a keyboard.
Others in this thread already pointed out that C# wants to go up while Db wants to go down. The reason for that is that for violins, cello, some flutes, and Organs, they are not the same notes. They just happen to be rounded onto the same key on most modern electronic keyboards.
Since you asked to learn about harmonics, I would therefore advise against focusing on 12-keys, because that would hide the underlying complexities from your view. Harmonics are in my opinion best studied on analogue instruments like a violin, where you can actually play a musical comma, as opposed to pretending it doesn't exist.