Hence the value of learning some theory. Scales always follow the same pattern of intervals between the notes. Major ones are easy, minor ones are just a tad more complicated. Modes aren't too difficult either. Then you can write triads on each scale note.
Next you number all the notes off in the scale from 1 to 7 (normally we use roman numerals I II III IV V VI VII). Then you quickly realise that chord sequence can be repeated in any key using those numbers. You've abstracted them away from any particular key.
Assuming the key in your example was C major, the chords would be I, VI, II, II#, V, V7, III, IV.
So let's say the piece sounded better in F major. You can now pick out the chords in the F major scale easily -
F, Dmin, Gmin, G, C, C7 Amin Bflat.
I VI II II#* V V7 III IV *raising the flat in Gmin by a semitone. It would be 'B natural' but what this means is the third of the chord is sharpened. You can do it other ways: II-nat3. (Can't do a natural sign here)
It gets you to where you want to be a lot quicker - maybe some upfront work but when you get into more complex stuff it pays off. Just a step further and you'll listen to a song and easily work out its chords by ear.