There are a couple of steps to get it down, but once you have it you'll always have it and it is the easiest way imo.
Learn the order of sharps and flats and how they relate to keys. There are a couple of mnemonics that you can look up on the web for the order, and then there are a few tricks to memorizing how they relate to the keys. Usually the first chord and the last chord of a song indicate the key. There are exceptions, but it is generally so.
Each major key as seven notes (a scale). For example, the key of C (no sharps, no flats) the major scale is spelled C D E F G A B. Each letter is also a scale degree -- just add numbers.
C D E F G A B
1 2 3 4 5 6 7
Let's say you want to transpose a song from the key of C to the key of F.
The chords in the original version are C F G. Turn those letters into numbers: C, 1; F, 4; G, 5. You may have heard folks talk about songs that are in a 1-4-5 progression (often using Roman numerals). A 1-4-5 progression (or 2-5-1 or 2-6-5-1 and so on) tells us what scale degrees and we can just plug those numbers into to any key.
Since your target key is F, spell out the F major scale and add numbers.
F G A Bb C D E
1 2 3 4 5 6 7
In the key of C, the C chord is 1; in the key of F, the 1 is F. Etc.
C F G (1 4 5)
F Bb C (1 4 5)
The chord progression in your transposed key of F is F Bb C.
As I said, there a couple of things you have to get down at first (order of sharps and flats and their relation to keys), but that's the only initial heavy lifting. After that, it's a matter of counting to seven. There are some situations where this doesn't always work (non-diatonic chords), but once you learn the basics you can then learn the exceptions.