Here were are talking about harmonization of a tune.
It could be any kind of melody - a chorale, a hymn, a chant, a folk song, a popular song, even your national anthem (or mine).
The first step is to decide what chords will you be comfortable of using?
Tonic, subdominant and dominant? Their inversions? Dominant seventh chord? Its inversions? Other seventh chords? Or something entirely different?
Then decide on the key. Look for key signatures and see if the melody stays in one key or modulates.
Then look at the notes on the beats of the melody. Not all the notes but just the main beats.
Think what chord could go with each note of the tune?
Look at scale degrees of the tune and compare them with the scale degrees of each chord you know.
My guess is that any kind of melody could be harmonized with just 3 simple chords - T, S, and D.
Then you have to avoid parallel fifths and octaves simply by letting the bass go to the opposite direction than the soprano (sometimes alto or tenor). When possible keep the common notes of two chords stationary.
Yes, it will be boring harmony. Yes, the bass will leap a lot.
But that's a start.
If you want to go fancy, add inversions. Then the bass will be smoother.
Whatever you do, I really recommend transposition.
Transpose all your exercises to various keys.
It will have a compound effect on your skills sooner than you realize.
Hope this helps.