Black plastic is a good start. Leave it for a year to kill any weed seeds. If you know your newspaper is printed with soy ink, It makes a pretty good weed shield for a year or so but won't build up temps to kill weed seeds as the black plastic does. It breaks down fairly quickly under a mulch layer . UV will break down your exposed plastic in a year or two.
Other sections, you could mow/chip or perhaps rototill depending what's there. Spread with overlapping newspaper, (stake with carved wooden sticks from your trimmings should take just a few strokes with a hatchet or knife) Mulch with the trimmings to protect the newspaper and keep it looking nicer. Plant with your chosen ground cover in holes cut through the news paper. That's not expensive, but takes a little labor.
Don't try to do it all at once unless you have a big budget and time which it seems you don't. Clear a section this way each year and follow on the next year. Develop a master plan and screening to keep your neighbors happy with how your lot appears. Probably a good idea to talk to them and get their support anyway.
Landscaping cloth is your next help. This is like the black plastic, but water can pass through. Lasts much longer than the newsprint will, decades if well mulched. Cut out holes where you want to plant perennials. Cover with a mulch to help protect soil temps. You'll get weeds in the mulch in a year or two after starting this, but they're easy to pull as they can't root deeply. You can also spread a pre-emergent in the early spring or late fall. to help control new seed sprouts. Check to find one that is safer for food gardening if you're going to use it in your food areas.
Run a dripline irrigation system so only the plants you want are getting supplemental irrigation. This helps control weeds a lot in my neck of the woods, but talk to local professionals for the best info.
Use natives for most of your landscaping and become comfortable with an informal look rather than manicured. creeping thyme would seem to be a good bet for ground cover and weed control. It's fairly flexible to climate. many varieties and many are culinary friendly though not all. You'll need a few different such dense choices for the different parts of the lot. Some for shade, some for high heat and sun and so on.
As your chosen ground cover spreads, cut new holes in the landscaping cloth for the ground cover to root in. Plan your irrigation for covering these expansions.
Rent a wood chipper once per year to deal with your trimmings. You can use this as a mulch, or to feed a compost system. As a mulch, it robs the soil of some nitrogen in the first few years but will return it over time as it continues to break down.
Once you've got it under control, you can then start augmenting it with more variety and scale you can manage.