One easy way (surely not the best one!) to harmonise a given bass and melody with full chords is to look if you can harmonise everything with three chords, an idea that goes back to the 18th century: the tonic, the dominant and the supertonic, and some simple rules: The tonic (I) starts and ends the piece and can lead to all three chord types; the supertonic (ii) precedes a dominant; a dominant (V) can lead to back to the supertonic or to the tonic. Supertonic and dominant can be used as seventh chords with the seventh being resolved by going one step down, thus the supertonic seventh always leads to the dominant, the dominant seventh always to the tonic. You can use the chords in root position or inversion; the final cadence should see the dominant and tonic in root position. If the music has left the key, simply use the three chords of the new key. See e.g. Daube's Musical Dilletante for more information on this method of harmonising a melody.
If you write out which notes belong to which triads and seventh chords, you will easily see which chords can fit to the given melody/bass-combination.
There are far better methods than this one, but hardly easier ones.