Its my understand that bermuda type grass cannot stand up to being given a broad leafed weed killer. Weeds, as well as the grass itself, will be badly affected. Huh? where did you get that piece of misinformation? There are very few herbicides that will even phase Bermuda, and Bermuda grass eats broad-leaf weed killers for dessert. Round Up will just make it angry and will take revenge.

Get a "Weed Hound" from one of your local big box stores. It is a little tool that makes pulling weeds a breeze, especially ones like dandelions that have taproots. If there are too many and you need to use chemicals, things get a little tougher. I see you are in Zone 8, but where exactly are you? What type of grass are you going to grow? If it is a warm season grass (bermuda, zoysia, centipede, etc.) this is the time to seed so you need to get to work on those weeds. If it is a cool season grass (KBG, fescue, rye, etc) you won't seed until fall. This makes your job easier as you have more time to get rid of the weeds before seeding. About any namebrand Broadleaf Weed Killer will kill dandelions. However, read the directions as not all types are compatible with all grass types. There wil also be seeding-after-use instructions (ie. how long to wait before seeding). Also heed the temperature recommendations for use, unless there isn't much other grass that can be harmed by applying in the heat. I like Weed-B-Gon Max and have had good luck with it. You can get it just about anywhere. Here is a link that might be useful: Weed-B-Gon Max

Some of the new broadleaf weed control products do a good job on non-grassy weeds... An important thing to do is get a soil test... You may have some sort of soil deficiency that is allowing those weeds to prosper.... Working on your soil health will help produce a wonderful yard... Last is watering habit.... Deep and infrequent watering tends to encourage Grass and the few deep rooted weeds while Frequent shallow watering encourages lots and lots of shallow rooted weeds... One comment... Some folks hate clover... Some folks love clover... It's a free source of Nitrogen that feeds your lawn... Historically, most grass seed was 25% clover so that you wouldn't have to fertilize... If you do get rid of it -- you will have to make up by adding that much more Fertilizer... Thanks John

Controlled burns can work, however you've got trees close by....and they can also create more problems with other weeds. Some seed simply need heat and or smoke to start germination.... Chemicals are not an option for me, personally around food. Your choices are slim right now - basically hand weed around the plants that are growing, and then mow the areas that don't have food directly in them (yet) if you can, is probably going to be the easiest. You will do fine, you will "get" this thing. Realize that it is simply going to take time, lots of time for you to manage this size of space. My thoughts now are to save what you can by doing what I mentioned above. In the fall, concentrate on a much smaller growing space. Lay cardboard down (it will take a LOT and if you can get free wood chips from tree trimmers, neighbors or anywhere, get them. Spread them on the cardboard and leave it all winter. Let it smother what is below. But for now it seems your ambitions were larger than the amount of time you'd like to dedicate to working in a garden. That is the balance you are going to have to figure out...what you offer the garden in maintenance vs. what you have. There are also spray options if you so choose to do that. You can mow an area now, spray it lightly with kill, leave it in the sun a few days-a week (most chemicals work with sunshine) and then go ahead and mulch REALLY heavily (especially with cardboard and wood chips). Next year plant sunflowers in that spot, the year after you can plant food. (sunflowers are used to clean up toxic areas). You will however always have to watch the area - weed killer is only going kill the current weeds, not the seeds for the next year (or from past years). MULCH will be your answer, and it will make some lovely soil too :)