Keeping weeds down is more than just an aesthetic issue.
Weeds consume space, water and soil nutrients needed by growing vegetable plants. They also act as vectors for insects and disease, as well as providing cover for mice and other pests. When they grow out of control, they can even block needed sunlight.
Weed control is an ongoing task, but one that is made easier by employing a battery of methods.
One of those methods begins before the first seed is planted. Proper soil preparation can help reduce the problem before it begins. Churning the soil by a good tilling prevents weeds from getting started and disrupts the growth cycle.
Seeds will remain, but putting down a weed control fabric over the topsoil can help. It blots out needed warmth and light and provides a physical barrier that makes it more difficult for them to rise. Mulches provide a natural cover that will help suppress weeds. Manure, bark chips, sawdust, leaves and other forms can block sunlight.
Laying down a pre-treatment, such as a pre-germinating herbicide can prevent weeds from ever developing that might already have seeds in the ground. Exercise care to get the proper one. You don't want to make the soil inhospitable for your new vegetable seeds and transplants.
Proper seed selection will help, as well. Some fast growing, shading vegetables will block sunlight that weeds need to develop. They're sometimes called 'smother crops'. As they spread out under the soil, they help crowd out the weed roots, thus providing double prevention. Planting them at the correct depth, with the right fertilizer and water, will give them a head start against weeds. Their action can sometimes even help over winter to suppress weed development.
When weeds do get started, it's best to try to remove them mechanically right away. That's more difficult because there is less to grab onto. But the larger they grow, the more their roots spread.
We often think of weeds as isolated plants, but an 'underground view' shows that they're more like a tree on its side. The 'branches' are just the weed part that sticks up above ground. The 'trunk' is the web of material that connects them all. Getting one just disconnects that 'branch' from the 'trunk'. But the smaller they are the less time they've had to branch out. Getting one helps prevent their spread.
Hoes, weeding forks and other tools can be a big help here.
But when all those methods still don't entirely eliminate weeds - as they probably won't - don't be afraid of using a chemical herbicide. Take care to find one that will take out the weed and not your vegetable plant. But once you do, they can be used safely and effectively. Chemistry has developed a great deal in the past 50 years and there are many that are actually less harmful to you and the environment than some natural methods.