I really have to go get some lunch, but I will just add that what's needed for transposition is a good knowledge of all the intervals.
As I've mentioned on other threads, I've been giving music lessons to the children of a good friend of mine (8-, 6- and 4-years-old) and
I've made some flashcards for them. (They haven't been using them, but I hope they will start again.)

The have questions like this on them: What is the fifth of C?
Answer: G is the fifth of C.

I'd like to get them started on the variants, e.g.,

What is the raised 4th of Db?

The raised fourth of Db is G.

Ideally, the answers should come like a shot. With me, a lot of them do, but for some I still need to pause to think. Constant repetition is the key.

I'm good at sixths and minor thirds because I always think what the relative major or minor could be when I look at the key signature of a song I want to play.

These things are definitely worth practicing.