While I'm sure there are plenty of extensive ways to get this garden started, I've recently encountered the same thing and didn't do much to get my garden on its way.
I pulled all the existing plants out - trying to get the roots and all. Then I tilled the ground and went through picking up any weeds or roots that I missed. After I got everything cleared, I put down a layer of top soil and worked it into the existing soil. Dig out some rows and you're ready to go!
Regular maintenance is the key. Like one person said, you could use Roundup to get rid of weeds, but I didn't want to wait WEEKS and I was afraid the weed killer being in the soil would effect new growth. Just get out there once a week and pull up any weeds you see popping through the ground.
If you're looking for a cheap way to start seeds, try a recycled egg carton! I was really surprised at how well mine has worked out for me :) I got a bunch of tomato seeds started and in the ground in no time.
It's probably a good idea to still check with the local nurseries to find out what kind of plants grow best in your area, as well as what pests to look out for.