There are two types of weeds based on their lifecycle, perennial weeds and annual weeds.
Image Credit: (cc-by-nc-nd) Bruno Monginoux
Perennial weeds with creeping root systems, like ground elder, or deep tap roots, like dandelions, need to be dug out and burned. Include even the smallest piece of root.
Annual weeds will gradually decrease over the years if you make sure that you never allow them to flower or seed and remove them as soon as you see them, or even better hoe to remove them before they reach the light.
Mulching will help considerably by cutting off the light from emerging weeds but this cannot be done until vegetable seedlings are through.