I can't quite tell from your picture, but it appears you made a raised bed for your veggies.* It would be good for you to do the same for your new garden so the planting area is level.
I'm not sure exactly if you want the weed killer for maintenance or to remove what is growing there.* For weed maintenance in the veggie garden I recommend one of these hoes.* No chemicals, long handle so you don't have to bend over and they sever the weeds at the soil line.* The colinear weeder in the pic on the left is good for small spaces.
A circle hoe is also excellent for small spaces.* They come in long and short handled versions.
For large areas I recommend a radiant heat weeder.* I think there is only one retailer in the US.
I don't use synthetic chemicals in my garden, but sometimes vinegar can be helpful.* It's non-selective and can kill anything it comes in contact with, so you have to be very careful where you use it.* If you use horticultural vinegar it's very strong as it's 15% to 20% acidic.* Household vinegar is 5% acidic and only works on young weeds.
One other thought that comes to mind from your picture.* I don't know what time of the day the picture was taken, but from the angle of the sun it appears to be at the end of the day.* It appears there are a couple of young trees in the right side of the picture.* There's a possibility that these trees could shade your garden for a significant part of the day once they mature.* Just something to keep in mind as a veggie garden should have at least 6 to 8 hours of direct sun.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.