Gardening and Horticulture - Weeds... in a garden.

I just planted my first garden. I 'tilled a spot that used to be grass. Our garden is roughly 20' x 25'. We raked and de-rocked the plot before making our rows and planting our plants.
The plants are doing OK, but I can see clover and other grasses coming back. Is there anything that I can buy to easily get rid of the grass and clover without any harm to the vegetable plants?

Well you can use a hoe or pull the weeds but if your lazy like me [I prefer to say I don't have enough time] you can cover the ground with newspaper and then mulch over top of it.

Leave the clover!! It fixes nitrogen to its roots and the plants use the nitrogen. This year, you may be skunked with weeds, but next year, taking Marksr's advice, heavy mulch will help eliminate the weeds, or make them so easy to pull. Now be sure to used "composted" mulch, as new mulch or chips will have weed seed in it, and you will have negated all your work.
Next year, sometime in March, dig up your garden, heavily mulch it and let it lay until planting time. By then all the grasses will have given up and you can plant.
Now, that is not to say you won't have weeds!!! You will, but they will be easier to deal with.

Look for products under the brand name Preen.They include weed control products that do not harm other plants.You may have to remove new growth but the Preen will control further growth.

Preen Vegetable Garden Weed Preventer | Preen.com (http://www.preen.com/products/preen-vegetable-garden-weed-preventer)

...mulch over top of it.

I see that "chandler" mentions "composted mulch." Even if I use Preen like suggested by "spdavid," would using some mulch be a good idea, or a bad one? I have been considering using mulch because my soil drains very quickly. I was hoping to retain moisture and also combat the weeds but didn't know which mulch to use.

I got a later start on my garden (been in two-weeks now) than my neighbor due to equipment issues. Any ideas on how to catch me up to him? It sucks seeing his garden flourishing while mine is floundering, still.

I've not used the Preen. Basically I just use whatever mulch I can get the cheapest:D
I put the mulch down over the paper, mostly to hold it in place. In the fall, I'll plow it under and it will be composted by spring.

Plenty of water and maybe some fertilizer is about the only way I know to speed up a garden.