This takes me back to 1978, when I was first learning how to play keyboard on a home organ. I meant to play an F-major chord, but hit a Bb instead of an A for the center note and discovered the Fsus4 chord by accident! And since then, I have also used sus4 and sus2 chords in a lot of my music, especially original compositions. You can also experiment with 7sus4. As you might expect, it is like a regular 7th chord, but based on a sus4 chord instead of a major chord. So, as an example, whereas a C7 chord is C-E-G-Bb, a C7sus4 is C-F-G-Bb.
As for the more esoteric chords, like 9th, 11th, 13th, and beyond with variations, as has been said, you can experiment with those and learn them at your own pace. When I started learning organ, my teacher gave me a chart with all of the notes for the basic major, minor, seventh, minor 7th, and possibly diminished chords (It's been over 45 years -- I don't remember all the details). Over the months and years, I just started getting interested in learning the less common chords.
In reality, it's based on a pretty basic stack of major and minor 3rds. Since they are essentially based on the 7th chord, which resolves from the fifth to the tonic, you start with the fifth note of the major scale for whatever key you're in. So, as an example, if you're in the major key of F, then you start with the fifth note of that scale, which is C. Remembering that the B is flatted in the key of F, you have this progression of thirds:
C - E - G - Bb - D - F - A
Now, yes, we started with the C, which is the 5th note of the F major scale. But when numbering the chords, it is based on counting notes from the first note of this sequence. So, the Bb is the 7th note (the 7th note of the F-major scale, but when counting from the C note), the D is the 9th note, and the A is the 13th note.
You already know that C-E-G is a C major chord, and that C-E-G-Bb is a C7. When you add the D, you get a C9. Add the F, and get a C11. And then add the A, and get C13. I believe in certain circumstances you could even add a high C for C15 and a high E for C17, but some will argue that, since you are repeating some of the notes, that those chords are redundant. That's a discussion for another time.
But going back to C9, C11, and C13. Often, just the top 4 notes of those chords are played, as long as the root note (the C, in this case) is being played by bass somehow -- either in an arranger's auto accompaniment, a separate bass player, or, with your left hand (while you play the other notes with your right hand). So, a C9 is E-G-Bb-D, a C11 is G-Bb-D-C, and a C13 is Bb-D-F-A, again, with the C being played in the bass. A little twist is that for the 13th chord, the 11th of that chord (the F) is often flatted to the 10th, so a C13th is often played as Bb-D-E-F. This gives it some dissonance and character.
And, another thing that may help is something you may already have noticed by looking at those chords above: Ignoring the bass note, one chord will often have the same exact notes as another chord in a different key. As an example, look at the C11: G-Bb-D-F. That is is the same as a Gm7, except there is a C in the bass with the C11. So, if you want, you can think of an 11th chord for a particular note as having the same top notes as the minor-7th for the note a fifth above it (as C11 has the same top 4 notes as Gm7, and G is a fifth above C). Just remember that you want to have the root bass note -- the C, in this case -- played somewhere, to give it the full sound of the 11th chord.
A C6 (which I didn't list but was mentioned in another post) is C-E-G-A, which is just an inversion of an Am7 chord (A-C-E-G) -- the difference being the note in the bass. For even more fun, put a D on the bass with those notes, and you get a D11!
And finally, like Mike said, other modifications are just spelled out in the chord. Gm7b5? Take your Gm7 (G-Bb-D-F) and flat the 5th, which is the D, for a chord that is: G-Bb-Db-F. C7#9? Just take your C9 chord (E-G-Bb-D, with C bass), and sharp the 9th: E-G-Bb-D#, with C bass. In some musical notation, the symbols for plus and minus are used instead of sharp and flat when describing these modified chords, such as C7+9 being the same as C7#9.
This is all quite basic, as there are nearly endless variations of these chords, but just like learning the basic major, minor, seventh, and minor-7th chords, as you practice and get used to these more complex chords, they get easier!
