First, a few things that I've discovered don't work so well for weed suppression:
Woven tarps. They let water through, and they tend to catch and pool water. The area underneath of them can turn into mud indefinitely. They may work for weed suppression if you're going to take them up a bit before planting? But in Maine, where I tried it, it didn't work at all.
Bioplastic. Maybe thicker grades are better? We tried a thinner grade at a farm I worked at and it all blew away or got so many holes that it didn't make a difference. Same thing with the purple paper product being sold for use on raised beds. The edges disntigrate and it all blows away.
Straw mulch helps, but I've never seen it last the whole season either.
Wood chips... after you've done a few layers, they do make the weeds much easier to pull, but they won't make an entire season.
What does work:
A thick layer of compost. It's very important that the compost be finished compost. If it's still hot it'll mess with seedlings big time. Compost on top of bioplastic or paper or cardboard is even better.
Silage tarping. Often you can get pieces free from dairy farmers. Make sure to weight it down well. If you irrigate or water with this your plants will be so happy, especially sun loving plants like tomatoes. The reflected light off the black surface is great.
Planting things tight together to shade out weeds can certainly be helpful too. Kale, brassicas, and squash work really well with that method. Interplanting can work, but it's all about timing, which can be incredibly tricky..