Off the top of my head:
Just keep writing. The more you write, and the more plays you've seen, the better. Read some of the minimalist guys, Pinter comes to mind - where the dialog is so sparse, its almost nonexistent at times.
You want the audience to be thinking and making up their own minds about your characters - so play with what you reckon your audience is 'seeing,' at times - and then break
their preconceptions. (For instance, purely as an example - if its a scene where it appears to be an old man and a young girl - how long can you hide the fact its father and daughter? DO you want to hide this fact? Its a game you can play as a writer constantly with your audience. It forces them to think a little, rather than sit and be spoonfed 'character information.'
Same kind of 'gameplaying' can happen throughout your script, where you make the audience think 'A' and then sideswipe 'em with 'B.'
Also, theatre isn't radio with pictures. If an audience can see something happening, there's no need to spell it out verbally. Ditto using characters names too much (if at all) - that's a big area where a lot of plays fall down, people keep using each other's names in dialog - when in reality, you may well chat with family or friend for an entire scene without using their name. Trust that an audience is as clever as you are and will stick with what you're doing and showing them. So create stuff for them as if they are your equals. If you wouldn't tolerate X, then why let your audience have to? Stage and audience are on equal footing - they're just watching a story unfold. Make it fun for yourself and them.
(I mention Pinter because of the fun he's having with rhythms and silences - even though his plays are deadly serious things. Same with Tom Stoppard. There's always a ghoulish sense of fun discernible behind the lines. You can 'feel' the writer playing with language, turning conversation into drum solo's and conflict, using syllables and pauses for rhythm and effect, like jazz improv.)
Also - think 'scenes' - where your characters discover (the situation they're in) or (their relationship to one another) - and more importantly, how soon you want the audience to know these things. Who everyone is, and/or 'why they are confined to the area on the stage in view of the audience.'
Sometimes its fun when the audience gets to feel 'more clever' than the characters. Or is cringing at the bad mistakes the character's making - perhaps in a social situation, etiquette etc. Theatre's wonderfully interactive in a true sense.
But step one is: have fun and just write the story, let the characters chatter and have the plot line interfere and overwhelm them and force them to respond to
it (whatever the 'conflict' is that's getting in the way of your characters lives). Don't let an absence of formal training get in the way of your self esteem - or your stories. I was never trained in Drama, but have made a living from my plays being staged for a long time. Its about your stories, not about 'you' or any pieces of paper saying you're now officially 'allowed to be a playwright.' The play's the thing.
Had an Ivy league university stage my stuff last year, got surrounded by people far more 'educated' than I was - but they couldn't write the story I could. Same can happen with you. So don't let the absence of formal qualifications get in the way of the act of simply 'writing the script.' The script is all that matters to those choosing what plays to stage, not where you come from or your degrees.
my two cents worth