Good question. Not sure I have a great answer. This is year #4 at this house w/gardens. Weed pressure is a constant battle. I've read a ton of literature about different ways to address it (confession...I know way more about vegetable gardening that I do about keeping my yard nice). THe silage tarp for a couple of months before the season is something that a lot of biointensive growers are doing it. This is my first year trying it, so I'll report back.
The two keys, though, I think, are 1)the daily maintenance. Get a good hoe to get weeds (like this stirrup hoe: http://www.johnnyseeds.com/tools-supplies/long-handled-tools/hoes/stirrup-hoes/5%22-stirrup-hoe-9500.html) and life will be easier. 2) I think having good enough soil to plant intensively/closer together helps a ton. Much like growing grass, I've found that by increasing plant density, weeds are pretty much a non-issue. I've started planting most things way closer than the package recommends. If you want to read about some of that sort of intensive garden, there are guys like Jean Martin Fortier, Curtis Stone, Elliot Coleman (the true pioneer) doing a lot of good work in this area. Curtis Stone has a great youtube channel (https://www.youtube.com/channel/UC-BlDCX__nCLs_ZF9meYQbw)