Weeds can be a mixed bag if they get out of hand and go to seed. So while they are still small, I try to at least top them and remove the green growth, but not fuss over getting all the roots out. I remove & compost the weeds provided they haven't gone to seed. If you keep topping most weeds before they flower - even many perennial weeds like dandelions with deep roots, they will eventually starve to the point that the roots no longer have stored food energy to create new vegetative shoots (there are a few weedy exceptions to this approach - like bindweed).
Don't forget, many weeds are edible too, so knowing the life cycles and benefits of the plants you are removing really helps. There's a good number of weeds I keep around because they offer something good, and deadhead after flowering or remove just before they go to seed so they don't spread to much. I try to spend no more than 1 hour a week weeding the whole front and back gardens.
I'm not a big fan of the gimmicky gardening tools, so I have two I use a lot. While standing, a good one for the type of weeding I'm talking about is called a stirrup hoe. They are really fast to use and cover a lot of ground quick. Something like:
http://www.johnnyseeds.com/p-5973-7-stirrup-hoe.aspx
While bending down lower I use a Japanese Hand Weeder (from Lee Valley) which is an awesome tool and super sharp and quick.
http://www.leevalley.com/en/garden/page.aspx?cat=2,2300,44822&p=62038
Once the weeds are gone, you could try using a thin layer of mulch to shade or cover the soil, preventing the weed seeds from getting sunlight + germinating and it will also reduce water loss. If you decide to do this, some basic points - make sure the mulch does not touch your desired plants at the base and no more than a few inches thick. I use hemlock mulch and mulch all my garden pathways too.