Welcome to Music: Practice & Theory!
I agree with most of what TopoMorto said in his answer and I wish only to give you some more information, and ideas, and some of my own experience as a tip.
- The melody
Almost always the melody is the vital part of any music. I guess you always have some tunes playing in your head! I know I do! Certainly that's where I start once I have a melody. I take the piano or guitar if I have them at hand and start playing the melody, finding those notes in my head and the intervals. Or I just start writing those notes in Guitar Pro if I have no instrument by my side. If my melody suits me then I leave it as it is, otherwise I start thinking about changing some of it in some places to make it more interesting.
- The harmony
After I've finished the melody (it could be how the lyrics should sound by the way) I start finding good and interesting chords. I play them in different ways in different variations. On a live instrument making up a melody and harmony usually goes together. Sometimes I start with the rhythm when I'm unsure what I wish to have in the end.
- The rhythm
Usually once I have a harmony and a melody I think of a rhythm. I think of how should the song go, how fast or slow, what should be stressed and what kind of percussion might there be.
- The arrangement
This is the most interesting part for me. When I have a basic harmony, rhythm and a melody ready I start arranging my piece. I take other instruments and play notes, chords, riffs and etc. to add elements to the song. I add second harmony sometimes and second rhythm and search for sounds and ideas that make the song interesting. After I've added some other instruments I usually relook at my previously played or scored parts and sometimes change them.
- The effects
This is optional, however, sometimes I like to use effects in my music. Often I start the whole idea of a song with an effect.
Basically it is all about a good melody and a nice arrangement. I often listen to different music that I like and analyse it. At first when I was learning to compose I mostly copyied my favourite bands and composers.
Scoring a piece isn't composing... you should already have the piece you wish to score. And it's usually simple once you know what each instrument should play. Basically in scoring the most difficult part is knowing how to write the notes and the elements on the claves and the staff. Knowing how to show the player what you want him to play. I mean you need not only to write the correct notes that you play but also the stresses you make and the effects such as tremolo and vibrato and fade in/fade out effects as well as many many others.
I'll give you an example of how I write music:
I start with a guitar or piano (those are my main isntruments). I start singing or humming a melody and just playing (improvising) and searching for an interesting chord progression if I have no idea in my head. If I have one (maybe a lyrics) I start singing the melody or the lyrics and start looking for a harmony all at the same time! Once I've found the basic harmony and the melody is okay. I start playing in different styles and adding extra chords and notes. I start by playing mostly simple chords and no arpeggios or whatever. When I have some good interesting chords I usually record that. After I've recorded I play it and listen... and continue improvising in my head. If I feel I've spotted an interesting moment.. I stop and play that part with the new idea. After I've made the piece look good for melody and one instrument harmony. I start playing a second instrument to the music and create an arrangement. I tap out a rhythm and think of a bass line and etc. During the whole process I might change my main instrument several times if somewhere some note is disturbing or doesn't sound right with other instruments. I often do all that for a very long time. Once I have most of the music written I start thinking about a solo and effect If i want. Usually a solo takes up a lot of time if I wish to make it really interesting and not just a simple improvisation. I might even change the harmony is some places according to a solo I'm building. In the end, finally, I get a nice good arrangement ready for my whole band to play. Most of my arrangements are in my storage in gp Guitar Pro 7 format.