Melody notes like to live in chord tones. Your melody is likely to be the root, third or fifth of a chord, so try to test out those combinations and see if you find one that is pleasing to you. Of course, not every note is going to be a chord tone, so you need to use your ear and the context to figure out which notes are and aren't emphasized harmonically.
Once you figure that out, you can start with a simple progression of tonic, subdominant, dominant and back to tonic. An example of this would be I-IV-V-I, or in C-major, C, F and G major chords. However if you want to make it more exciting, you can feel free to use all of the diatonic chords available to you in the scale and if you want to be really adventurous, start working with other harmonic devices which a teacher can show you!