Learning music has to be rigid. If you skip the fundamentals, nothing else that builds on top of those will ever work. Doesn't matter if it's "just for fun" or a full concert at the Proms. If you want to swim just for fun, you better learn the fundamentals, or you'll die.
Back to your original issue. You already work entirely using the letter names of notes, so you already know how to transpose chords. Chord symbols are exactly the same as letter note names plus a few extra symbols. So if you can transpose the letter C to the letter D, you can transpose the chord Cmin7b5 do Dmin7b5. Same exact transposition. As for the individual chord tones, think in scale degrees. A major chord is 1 3 5, i.e., a C major chord is C (1st note of scale) E (3rd note of scale) G (5th note of scale). So a D major chord would be D F# A. See, you're not transposing every note. You're building the chord in the new key using scale degrees. This one concept will simplify your life immensely.
All of the above builds upon fundamentals:
1) knowing first 7 letters of the alphabet - ABCDEFG
2) knowing intervals- C to D is a whole step, C to C# is a half step, etc.
3) knowing your major scales - notes 1 3 5 of C major are C E G
You probably know 1) and 3). I'm not convinced you're solid on 2) yet, so with that little keyboard you have, try to master that, and you'll have all the tools you need to transpose individual notes and chord symbols.
I know you hate when I do this, but here's a little test of your interval knowledge. Fill in the "?"s below.
Transpose the following up one whole step:
C to ?
B to ?
E to ?
Eb to ?
Bb to ?
C# to ?
G# to ?