First- you need tools. A hand cultivator, which is like a long, bent fork; a trowel, which is a tiny shovel with a short handle that. You could choose a hand pick, which is, as the name says, a hand tool with a pointed end and a blunt end. Use the cultivator or the pick to pull the entire weed plant out of the ground; use the trowel to get up plants that don't cooperate.
If at all possible, weed when the ground is moist (not soaking wet) so that the roots come up easier. If the ground is bone dry, weeding is horribly frustrating.
Most weeding must be done on hands and knees - you get better torque that way.
Once they're all up, mulch away. If you're mulching around shrubs, trees, or any other plants, the mulch should be just up to but not touching the plant itself, especially woody plants.
You can compost the pulled weeds, but some truly pesky weeds - dandelion, pokeweed, dock, onion grass - can and will regrow in your compost bin. For that same reason, don't just leave them to be mulched over - they could regrow. Ditto on throwing them in the woods - they'll regrow there too. Compost what you can, and anything that looks like trouble, put in the garbage.
As far as mulch goes, you can get free wood chips from most tree companies, and they'll deliver a huge pile right into your driveway. Yeah, it's hard work to get them all where they need to go, but free is free.

Most of the people on Gardenweb are great: helpful and encouraging to everyone. Notwithstanding this, I recommend that you get a garden encyclopedia-type book. Reader's Digest has one, Better Homes and Gardens has one, and there are probably others. Such a book will help you to learn the basics, and how to ask the right questions.

I wish you all the best luck as you discover gardening - have fun!

Thanks very much! That was great!!

I don't really know how to identify those extra-pesky weeds, so I guess I'll toss all in garbage.

Can you recommend a type of rake for distributing the wood chips if I do opt for the driveway dump? I don't currently own a rake.

More tools! You will find as your gardening addiction takes off that you really, truly, need 3 shovels, 2 pitchforks, at least 4 pruning tools, a wheelbarrow, a hoe...oh, the list is never ending. The good thing is that you don't need them all at once.
If you get the driveway dump, you will need a wheelbarrow and some long-handled pulling or digging tool - a shovel, spade, garden fork, cultivator - to get the chips from the pile into the wheelbarrow. The issue with wood chips, free or boughten, is that they are irregular in shape and size, and tend to clump up; if they are the free ones, there are probably some branches mixed in with the chips, and the branches stop your shovel dead as you're trying to get a load of chips. I use a shovel (the kind with the slightly pointed blade) to get the chips out of the pile and a hand pick to reach in and loosen up the pile, to make it easier to get them on the shovel.

You won't need a rake for this - gravity does most of the work for you, when you upend the wheelbarrow where you want the chips to go. Then you get down and shove them around a bit with your hands (on which you have gloves!). If you have an area that is hard to get at with the wheelbarrow, like behind a bush or between two plants, you can put the wood chips in a bucket of some kind and use that to control where they go; I use the buckets that cat litter comes in, as they're easily dumpable.
You want the mulch to be 3 to 4 inches deep, which will slow down or stop most weeds.
It sound like a lot of work and it is, but it's fun anyway, moving all that fresh wood around, getting up close to your bushes so you can see how they're doing, and knowing that you are improving the soil, making your life easier, and pleasing the neighbors, who will then realize that they, too, should be mulching. And when they're moving wood chips around in August, you'll be sitting on the porch!

Lisanti, one problem is no garage. Or tool shed. Or anything. I do have a basement with a door opening up to the middle of the back lawn, but that's seriously downhill from the front lawn (which comes up to my front door at ground level).

I'm thinking of hiring a landscaper to do the mulching....though I guess I'd better do the weeding, because nobody will be as thorough as I am. But if I change my mind (like if the weeding gets me addicted to the clean - cough, cough, cough - air and all), your tips will be invaluable. And I'll probably mulch the smaller plots myself.

Hmm, one issue I just thought of. That front bed I was originally talking about, full of terrifying jungle weeds, slopes severely downward toward the street. I probably can't make mulch stick to the slope. Should I just keep using a machete or weed wacker on it?

Sorry if I sound hopeless. As a lifelong NYC apartment dweller, I'm sort of like Zsa-Zsa in Green Acres.

Keep in mind that those "weeds" (a "weed" is a plant you do not want growing where you do not want it to grow) have been taking nutrients from the soil they have been growing in and if you throw them away you also throw those nutrients away. It may also be illegal for you to put yard waste into your trash,something you need to check.
Those "weeds" can be uprooted and layed back on the soil to be, eventually, digested by the soil bacteria which will recycle the nutrients those plants have back to the plants you want to grow. If the planting beds are not now mulched you may well want to do that, but you need not spend your money buying something at your local garden center. In New York there are trees that every year drop their leaves which are a valuable resource even though most people think of them as trash that needs to be cleaned up and thrown out. Those leaves can be used as mulch material. Newspaper, layed on the soil will exclude any plant covered of its access to sunlight and that plant will die, can be put down and then covered by things such as leaves to hide the paper and to hold the paper in place. The newspaper will also help extend any mulch material you have, help it cover more area since the mulch can be thinner then what would be needed without the paper.
Mulches will often stay in place on slopes unless there is a downpour and the rain comes too fast, too soon. Light rains do not move mulches, although they can and do move soil and that is called erosion.

"don't just leave them to be mulched over - they could regrow "

but...

"Those "weeds" can be uprooted and layed back on the soil"

So....which is it? Not trying to start a gardening flame war, I just don't know what to do.

The leaf/newspaper trick is clever, but not viable in June, unfortunately.

As you can see, there is no one right way to do anything.

Your questions regarding the slope, your lawn (in the other post), tool storage,and the weeds have led me to a deluge of other questions, which leads me in turn to recommend that you start another thread in either the Soil, Compost & Mulch or the New To Gardening Forum, both of which have a lot more action than this one. This is some of what I, at least, need to know to help:
Where are you? Not just the state, but the area.

How much rain & snow does your area usually get (you can guess)?

Is your yard shady? Full sun?

Is your house new or new-ish?

How steep is the slope? Can you guess at the angle? Is there anything at the bottom (landscape timers, rocks)?

What is planted in the sloping area already? Is it healthy?

What do you plan to do with the area in the long run?

Pictures would be extremely helpful.

Lisanti, thanks yet again. Thing is, I feel so comfortable here in Weeds. It's like a hip bar after hours. I can stretch out and relax away from all the noise and smoke. I feel like Weeds is really me.

Rain and snow: I'm just north of NYC. So decent rain, and wildly varying snow. Based on last winter, we have the snowfall of Fish, Alaska. This winter, unsure.

Yard is full sun in middle, shade on the edges.

House is 100 years old. Need some asbestos? Perhaps some lead paint chips?

I can't guess the slope angle, but let me go out and shoot some photos. Currently in bathrobe, but don't want to do a Vincent "The Chin" Gigante, so it'll have to wait for after breakfast.

There is almost nothing healthy in sloping angle, except some low green pachysandra-like crawler I can't quite identify, but which the aging alcoholic dude who mows my lawn supposes - and I agree - to be intentional. I'll try to include that in the photo.

The aging mowing alcoholic also says I should shovel and turn the soil on this slope; he says it's my only hope for removing all the weeds. He grants that I may break off some weeds at the roots in the process, but insists that, even after rain, I'd do the same to some of them pulling them up by hand.

In the long run: I plan to have this look really really good without the slightest effort.

Photos to follow as soon as I work through the rest of this oatmeal.

Think I should just hire a gardener? Are any of you guys available?

Been too rainy for photos. By the time the rain stops, the weeds will be up to the phone wires. Stay tuned.