piano transition from d major to c sharp?

Sure, here's what I used to do as a church organist. It works in rock, or anything else.

From your full tonic chord in D major, drop your bass note to Bb with harmony filling out a Bb7 chord with the same D from the D major chord voiced on top. Resolve that to an Eb chord with Eb voiced on top and also in the bass. QUICKLY add the b7 to the Eb chord making it an Eb7, keeping the same top and bottom notes. Then drop the bass note to Ab (enharmonic to G#) with the right hand harmony playing the Ab (G#) chord, still voicing Eb (D#) on top. Adding the b7 to that is optional; I wouldn't do it, though. Finally, resolve to C# major with the root in the bass. Let it all happen in no more than a couple measures, and it will sound quite right for you, particularly with a rhythm pointing toward where you want to go.

Good luck.

I'm a bit out of the loop on harmony, so don't laugh.

D M - B flat M - A dim7 - A flat 7 - D flat.

The A dim 7 to A flat 7 just means dropping the A to A flat. You then have the dominant 7th of D flat, which is enharmonically the same as C sharp. It sounds a bit sudden to me, and someone will probably give you a longer and better way. But as I said, see my disclaimer above.

Such "semitone" chord progression are usually used in modern pop music when the "climax" of the song comes, the same boring chorus part is transposed up 1 semitone.

Usual old trick by many modern composers that doesn't require any transition chord(s) in-between.