In all my years as a pianist in opera theatres, transposing was part of my bread and butter. First things first: there is no quick fix. There is also no standard method: depending on your musical training (movable Do solfège, fixed Do solfège, jazz, classical...) you will find different ways to transpose. Here's how I'd start learning it:
- Take a piece you know very well, not just "half decently".
- Play through a phrase of it in the original key, then see if you can play it in a neighbouring key, no more than a tone up or down.
- Try to transpose whole chords or fragments of a melody in one go. It's like sight-reading: if you do it note by note, it will take forever. You need to understand harmonic or melodic elements and transpose each one as a chunk.
- Little by little, you'll get better at it. The better your grasp of a piece in terms of harmony and scale degrees, the easier it will be.
If you decide to learn a song that you really want to sing, and you need to transpose to a distant key, and you aren't already proficient in transposing, then it'll be easier to do it like this: enter the whole song into a notation software, then ask the software to do the transposing for you.