I use no chemicals. I eat dandelion leaves for one, and I would not want to eat them if they were covered in chemicals. That is quite frugal. =-)
We have zoysia grass, which is the perfect grass for the frugal. It grows slowly, looks like a carpet, and grows so closely that it leaves no room for weeds. I have a great lawn, which I never water and never weed. Yay!
http://www1.zoysiafarms.com/At the start of every summer, I weed the flower beds. The veggie gardens hardly get weeds (will tell you why in a second). After I'm done weeding--and, boy, are there some invasive plants out there!--I cover with mulch and rocks we gathered from the beach (free decoration, which I can return to the Earth when I die!).
The mulch, as you probably know, smothers the weeds. Some will still come up, so you'll have to pluck them out. When you pull, try not to scatter seeds around. There's this one plant that is designed to scatter seeds EVERYWHERE when you pull it out. Sigh. You can't do much about that one. That weed was terrible last year, but another weed is taking top billing this year.
For the veggie garden, I plant lettuce--for instance--close together. When the plants grow to full size, they lightly touch each other. This helps the soil retain moisture while not allowing light to the weeds that might grow below.
You could try putting newspaper down instead of netting and then put mulch on top of that. I do that in the veggie garden, and it works well.
Annuals generally do not come back. If you see plants coming back, they are perennials. =-)
Except....I have two "magic" annuals. They are not supposed to come back again and they do. It's wonderful. One is an impatiens and the other is a daisy. Remarkable plants.
Plants that are bulbs often need to be thinned. You can sometimes find people giving away these plants on freecycle. I recommend checking that out. Just be sure you're not getting an invasive you don't want. Mint, for instance, puts out runners. I am happy to have mint grow anywhere it wants, but that would drive other people insane. Greek oregano is another great plant. Smells great, is edible, and will grow most anywhere.