How do I kill weeds in a way that won't hurt seeds and plants?

Never spray weed killer on plants. You need to dig out the weeds by hand, or smother them with newspaper, cardboard or landscape fabric. No sun = no grow.

Pull the weeds would be the safest for your new plants. Make sure you gather any seed heads left from last year, and/or pull new weeds before they go to seed.

Don't use the spray on the weeds, if any gets on the plants you want to keep it will kill them. Pull the weeds by hand, making sure that you get the roots. I use Preen Weed Preventer on all my gardens and it really works well on seed born weeds, which I assume your weeds are from. No matter what you do, weed seeds will end up in the garden from being blown in by the wind or dropped by birds. The Preen keeps my in ground gardens free of everything but root bound weeds like creeping charlie and some grasses. Make sure you use some kind of mulch, I use shredded leaves to cover mine in the fall and spring and then switch to grass when we start mowing the lawn. Preen does not affect any seeds you have planted or the existing plants in your raised beds. Good luck with your weed problem, Dee, you can easily get rid of them!

All plants need light, including weeds. Use a pringles can or any other can. Campbell's Soup cans work too. Put it over the weed, if you can turn and push it until the can is 3 inches into the soil. The Weed will die.