1. Non-glossy newspaper
3. Hay or straw
4. Shredded paper
To make it weed free longer you need to use thicker stuff. I use newspaper six pages thick and over lapped, then I cover it with shredded paper or hay to hold it down. (Note, if doing the mulching on a windy day you will have this stuff all over your yard. I know trust me.) When going around the plants I rip good size tears in the paper so it can get close, but not damage the plant.
Be careful around plants
I forgot to mention I run a soaker hose under the newspaper when I make it this thick. It tends to block rain from getting through for a while too. Of course I didn't think to take a picture of this. If you don't have a soaker hose make a ring of dirt about 3 to 4 inches away from and around the plant, and do not cover this up with newspaper. Instead water the plant in this "water holder", it acts like a bowl to hold and collect the water. You can put hay or shredded paper in it for some weed protection.
Shredded paper around Horehound, see the holes where weeds have gotten through?
I use cardboard for walkways or for around runner plants, like cucumbers and watermelons. I also use it around squash plants to help with weeds. Basically anything that is a spreading plant and needs room weed free I use cardboard. I never put it up directly at the base though because it blocks water. It is perfect for walkways and areas you don't want weeds. I don't get to use it much since I have a small garden at home, but I do get to use it at the cub scout garden and it works great.
Topping it off
When you get the newspaper or cardboard down wet it well. This helps hold it down and even in a little wind this is very helpful. Then I put my topping on, the shredded paper or hay/straw. When I get this done I wet it again, that stuff likes to blow away even more than the newspaper. I also continue to wet it down a couple of time a week until it sticks to itself. It normally takes a few weeks for this to happen. After that few weeks and things sticking together, you might notice a few holes in your system. Mostly because you will see a few weeds.
Straw that has been patched with shredded paper. It looks weedy, but some of it is garlic.
Also note the soaker hose end, don't forget to leave an access point.
Don't worry though, its an easy fix. I cut or pull out the weed, fix the hole by moving around the newspaper, or add more, and recover with more top mulch. Rewet again as needed in new spots. This usually works well into the summer, until the garden is so big and its so hot that it looks like a jungle. That's about when the mulch is really decomposed by weather and bugs. You will notice a lot of bug life if you peek under the paper. This method is also how I put my garden to bed in the fall. At least that's what I call it. Only I don't leave a soaker hose under it and there are no plants. It encourages worms to stick around and lets them go higher in the soil than if they were out in the yard normally. I even throw some scraps under there sometimes, just in case the worms get hungry. You can leave this down until decomposed and turn it under in the spring, it doesn't hurt. I always notice a darker, richer soil when I keep it on and turn it under the next planting season. Once again I will say you can never be too careful about the hay or straw, where ever you buy it make sure and ask if pesticides or herbicides where used. If you can't afford news papers ask your friends or neighbors what they do with their old ones. Its a great way to recycle. I also look for cardboard at places of business, they are happy to get rid of it. How do you mulch?