Hand weeding - little and often - followed by mulching. Hand weeding also means you become very familiar with your soil and learn to id anything growing in your garden.

Plastic is not a good idea in an organic garden for several reasons starting with limiting the exchange of air in the soil.

Newspaper or cardboard works just as well at cutting off access to sunlight those plants need to grow and also contributes some organic matter to the soil that plastic will not. Hand weeding is also a good method as pointed out by floral, but any mulch material used, shredded leaves, wood chips, shredded bark needs to be thick enough (3 inches minimum if no other sunlight blocker is used) to block a plants access to sunlight.

I hand weed at least 15 min a day. My garden is small, about 6 beds, 400 sq/ft. I make a a circuit through all the beds every 6 or 7 days. By the time I work around to the beds I weeded a week prior, new weeds are back up, like bind weed or dandelion. I don't mulch because in CO I want to get the ground warmer. I will mulch the cool season veggies starting in July.