Songwriting is among the best arts. People write in all different kinds of ways, this is just a way to start, if you are more comfertable writing a different way, go for it! (I only write for piano)
There are two aspects of any song, the lyrics, and the music. There is much debate as to which should be written first, however, I usually write the music first, so I'll start with that.
1. Come up with a good chord progression
I usually do this by messing around on the piano and improvising. It is good to knwo some basic theory for this. There are certain chords that always sound good together. Here are some chord progressions to start with:
aminor>Fmajor>Cmajor>Gmajor
Cmajor>Gmajor>Fmajor>Fmajor
Cmajor>Fmajor>Gmajor
2. Come up with a melody
I do this one of two ways. Either I'll play the chord progression and sing over it until I find a melody that I like, or I'll play the chord progression in my left hand and improvise over it with my right hand until a melody develops.
3. Figure out the style
Do you want a swing base, light sound, full sound, fast paced slow paced, whatever you want, but you gotta think about it!
(You need a different chord progression, melody, and style for the verse and the chorus)
Lyrics:
1. Figure out what you want the song to be about
A good way to do this is write a paragraph about what you're thinking for the song. You should know where you are going with it.
2. Come up with a lyrical hook
This is the phrase in the song that everyone remembers. Hooks are usually the chorus, and they've got to be catchy. Here are some famous hooks: "Let it Be," "Light my Fire," "Dance, Dance," "Hard Day's Night"
3. Write the chorus
The chorus should be catchy, and it should be built around the hook.
4. Write the verses
Consider taking from the paragraph you wrote about the song. THIS IS A FIRST DRAFT! don't be afraid to write because you think it has to be perfect, you'll go back and revise.
5. Write second draft
6. Revise again
7. Revise as many times as you need to
Now fit the music to the lyrics. I always craft my lyrics with the music in mind. For example, based on the melody, I know approxiametly how many syllables per line I am aiming for.
ENJOY!