So here are the two methods:
1. Move the notes to a different place of the stave but keep the clef the same.
2. Keep the notes on the same place of the stave but change the clef.
The first method is very simple - you just have to figure out the scale degrees of the original key and apply them to the new key (with new accidentals). The problem is when the music is chromatic or when it involves modulation.
Then you either have to switch constantly to the new key when transposing (which is really difficult) or you can think in the old key but with raised or lowered scale degrees of the first key (which in a way is cheating).
The second method works this way - by keeping the notes on the same place of the stave (with new accidentals), you have to figure out which clef would work for each particular instance. By the way, there are 10 clefs total (2 G clefs, 3 F clefs, and 5 C clefs). Each of them has a special name, like the treble clef, the bass clef, the soprano clef, the alto or the tenor clef and so on.
Sometimes people like to transpose by interval. In other words, if the destination key is a whole step upward, then they play the given melody but in their mind they always have to think about playing a whole step up.
Transposing by intervals is not nearly as good as the other two methods because it doesn't provide the opportunity to think in a different key. It may seem as a simplier solution to someone who is just starting but in the long run - I highly recommend the first two methods.
Because the first method has it's own issues, I personally use the clef transposition. Sure it takes time to develop this skill. Just remember how difficult the bass clef seemed to you at first when you only knew the treble clef. But after about 3 months any new clef becomes easy to use. So it's just a matter of patient practice.
But in the long run, knowing how to read the different clefs is an essential skill to have to anyone who wants to learn organ improvisation because anytime you want to use your musical theme in a new key, you can simply switch the clef.
By the way, I teach both methods of transposition in my Transposition for Organists Level 1 course.