Black polyethylene plastic sheeting is the most economical and probably the most effective weed barrier. 4 mil will last for one or possibly two seasons. 6 mil costs more, but will last 3 to 5 years here in Michigan (probably less in climates with extreme summer heat and UV light).

You need to poke or drill drainage holes in it. Once that's done and it's put down, secured really well against the wind, it's pretty much maintenance free.

Lots of folks use cardboard and/or multiple layers of newspaper under and around their beds. I built a couple new raised beds this year and I'm trying those methods.

I also have had pretty good success with mulching my veggies with shredded paper and/or shredded leaves.