I can already figure out the melody of a song by 'trial and error' along the C major scale just because it is easier on that scale and eventually become okay at guessing melody. This is fun enough on the recorder but I couldn't figure out how do I make a full use of string instrument such as ukulele. To sum everything up, I only have experience on playing one note at a time and if I try to play a song by ear I would pick up the easily distinguishable notes of the song. I don't know much about music theory other than reading a little bit of music notation, scales and notes that build a triad chord and everything is self taught. I just want to make those melody sound fuller on ukulele. Are there any way to make melody to sound interesting without referring to music score?
As far as having to refer to a music score, you don't have to, but that really depends on your creativity, and your musical ability. Some people can cook without a recipe. It will be very helpful if you were to find a chord chart for the songs that you're playing. That way you can look at the pdf that I sent you and find out how to play the chords that you need for the song.
I would suggest getting familiar with basic chords, such as all of the major chords, and all of the minor chords. Extended chords such as 7ths and 9ths, or diminished and augmented chords can wait until you're familiar with major and minor triads.
If you want to do it yourself without using a chord chart for the song, you can pick chords to accompany your melody that share the same note as the melody. For starters, I would suggest picking a chord whose root note matches the note that you're playing. Obviously this is not the only or best way to harmonize a melody, but it can be simple enough and with major and minor chords you shouldn't run into too many harmonic issues and it could sound pretty decent. It will also help if you can identify the key, because if you're playing a Cminor chord for a song that is predominantly in Cmajor, you can have some serious issues.
Basic Music Theory has a lot of helpful information. If you can identify all of the notes in the song, you can look at the scales there and identify what scale your notes belong to. Most songs however, do not stay in one key, and it may prove difficult to grasp at first.
At any rate, good luck, and use the chord chart! It'll be very helpful.
One thing that really opened my eyes musically was the concept of "chord of the moment". I had a teacher that mentioned it in passing once, and it blew my mind that "real musicians" can change chords that fast!
Basically, as a song is played through, musicians are always changing chords, and potentially, the key of the song which changes the chord family. Each melody note sounds good when played over the correct "chord of the moment", or saying it in reverse, given a specific chord of the moment, basically anything within that chord can be used to progress the melody. There's no rule that says that ONLY those notes sound good, but as a general roadmap its a good place to start.
Another eye opener was understanding that chords within a key are (again, generally) made up of the notes within key's scale.
One time I asked "how to I know when a key change would sound good, and how do I know what key to change to". That really blew my mind.