There is lots of good advice here from erudite people who care about this subject a lot. It is worth checking out older threads on this subject, as there is plenty of good stuff from fine writers like Jerry Rasmussen, McGrath of Harlow and El Greko to name but three.
I have a bit of experience of being on TV and radio and have played a few gigs along the way. I look for certain things when a new song starts to emerge.
The first quality I want is resonance. I tend to build songs around one line, which sums up the essence of what the song is about. This line can be the first line of the verse, the last line of the the verse, a line of the chorus or even (if the song has one) a part of the bridge. It has to sound natural, be easy to sing and stand up to repetition. Most of the best songs you know have a line like that.
So I start off with this line and say, "Wouldn't it be nice if there was a song which went…." I then try and listen in my mind to those lines which most naturally go with that first line. When I have that shape, the crossing out and rewriting begins.
From my days as a writing student, one of the most important pieces of advice I heard was, "Don't tell it – show it." Off the top of my head, a good example is Dylan's "Tangled Up In Blue":
"I was standing at the side of the road
Rain falling on my shoes".
You know it was raining heavily, because the narrator had to look down at his shoes rather than up at the sky. Great writers have an innocuous way of showing you more. The first verse of "Eleanor Rigby" has that quality too. McCartney does not tell you she is lonely – he shows you graphically.
Another tip I picked up from learning to write drama, was that if you have characters, never put words into their mouths. Listen to what they are saying and write that. Your characters should speak in their words – not yours! Jerry Rasmussen wrote a very good post on the importance of listening on one of these threads.
If I come out with something which sounds natural and which I could imagine someone else saying or singing, I am usually happier with the results. I try to leave lines uncluttered by extra syllables. "The man I met yesterday" is usually easier to sing than "The man that I met yesterday". It is easier to sing less syllables and it gives you more flexibility with your phrasing. If your lyrics are good they will also be easier to learn.
My final piece of advice is that you should never ever write twee songs about your feelings like Graham Nash does. This is because they are boring, only Graham Nash fans will listen to them and if you do, I shall come and shoot your cat!
Good luck with your new craft.