If you have a garden of any kind you are going to have to weed it. There is just no way around it. Mulch may slow the weeds down or make it easier to remove them, but the best way to keep a garden free of weeds and looking tidy is to perform this task FREQUENTLY so that they don’t get a chance to take over.
The best thing to do is choose one day every week to carefully inspect your garden and pull out or chop down any weeds that you see with your hoe, trowel, or weeding tool. You don’t even have to be super thorough about cutting out every bit of root. The benefit of doing this frequently is that the weeds never get a chance to grow very big. If you simply cut them down while they are still small, they will usually die. The ones that don’t die will have a hard time growing back if you repeat the process every few days.
Eventually, a garden that is weeded frequently in this manner just won’t have many weeds. There will always be some. Seeds will still make their way in from outside the garden, but weeds that come up inside the garden will never be allowed to grow long enough to flower and drop seeds of their own, ensuring that your weekly task of weeding the garden remains manageable.