I think you'd get different answers depending on who you ask, so this may yet be closed as being too subjective.
Even the same composer may start at a different 'point' depending on the ideas they have at that moment in time.
For example, if they heard someone whistling an interesting melody in the street that day, then they may develop that melody first before harmonising it and then orchestrating it.
Alternatively they may have come across an interesting chord progression whilst at the piano and later try to craft a melody on top of it.
In the world of songwriting, Elton John and Bernie Taupin certainly worked in a very particular way (from Wikipedia):
The 1991 film documentary Two Rooms described the John/Taupin writing style, which involves Taupin writing the lyrics on his own and John then putting them to music, with no further interaction between the two.
But that doesn't mean words-then-music is the "correct" songwriting method. Alan Menken/Howard Ashman (of Disney fame) took a different yet equally successful approach (from theartsdesk.com):
When Howard and I wrote “Be Our Guest”, I said, “You know I’m just going to give you some simple French piece of music.” [He sings the melody.] It basically tumbled out as fast as my hands could play it. “That’s the form, go take it, write the lyric, when you come back I’ll write the real thing.”
In terms of more instrumental composition, Howard Goodall (best known for numerous British TV themes) describes his creative process (pdf) when commissioned to compose a Requiem. A different approach yet again!
There are many "right" ways to go about composing music, but that's not to say there aren't "wrong" ways too. How you go about composing something will depend on what it is you are composing and what you have to work with. Some good questions to ask youself:
Is it functional?
If you are writing music to serve a purpose - say, a song important to the story in a musical or a jingle for an advert - then the form and content will be heavily dictated to you. Therefore, your focus should be on the best way to communicate that message/text through your music.
Do I already have a melody/text/chord progression?
If you happen to already have some idea of what your composition will sound like then it makes sense to develop it from that point outwards. If you have a melody, there'd be no point in writing anything else without that melody in mind; from the harmonisation through to the rhythms and instrumentation.
Does it need to fit a certain genre or style?
Again, this gives you something to work with - if you're composing a Romantic-style orchestral work you know you should be including some more exotic harmonies and wide-ranging dynamics than you would for, say, a Status Quo tribute band.
Having written all that, I can sum it up as "it depends", which probably means this will be closed!
