I take it that you want to remove the sunken areas that were flower beds, and make the whole area lawn? First I would try to do something about the weeds, especially if they are perennial kinds. Selective weed sprays should be effective enough, as long as you follow the bottle directions carefully. Rotovating them will just compound the problem, multiplying the weeds, and making them deeper rooted.
Once the weeds are reduced drastically, look for deals on topsoil. Signs on the road, "topsoil for sale"--or, rarely, "topsoil for free"--around construction sites, etc., can be a good thing, but you want to see what you are getting, and you want to know something of it's history. Rocks and construction debris make all gardening hard, even lawns. Soil full of noxious weeds, oil, or long-lasting weed killers are a garden disaster! If it's good, find out how much you need to fill the sloughs, and see if you can trade favors with a friend with a pickup truck to bring it home.
Once it's home, and piled in your driveway, take a spade and peel up the sod in the former flower beds, set it aside, and keep it moist. Mostly fill the low areas with the new soil, leaving enough space to return the sod that was removed, plus a cm or so for settling. Replace the sod, and gently tread it down, shuffling back and forth in one direction, and then perpendicular to that--or rent a roller, if you can. Ideally from sod cutting to first watering is less than two days--sod has no shelf life to speak of! Water well afterward, and possibly as often as every day for the next week or two, if the weather is warm and dry. Gradually water less often, until it's on its own. Frequent mowing, and regular feeding will help keep the weeds down. Cheap composted manure from the garden center, applied twice a year over the whole lawn will gradually homogenize the appearance of it, for the better.

There is a school of thought that reckons that if you mow ANYTHING regularly, you can turn it into lawn. If lawn is what you want, I'd level the flowerbeds and any other low spots as Tug suggests. Then just mow every week and feed lightly when it's settled down after being levelled.

You should be aware however that a good quality lawn is not a low maintenance option. If you want as little ongoing work as possible, shrubs are what you need, with low maintenance hardy perennials in front of them. Once they are established - watered if necessary for the first year - they will need hardly any attention.