The goats might be fun, but they wouldn't be any more effective than mowing in terms of preventing things from reseeding. If your big problem is reseeding, you can put down pre-emergent herbicide to prevent germination--this won't do anything to control growth of weeds that are already there, but if you reapply as directed it can prevent new plants from forming. The catch is it prevents all seed germination not just the weeds, so if you have wildflowers or something that you were trying to grow there and were hoping it would reseed you'll prevent those from coming up too.
caitlin, the Jimson Weed we have here is very hard to get rid of, and it makes zillions of seed. If you try to dig it up you have to get every last bit of root or it will just start a new plant... What works for me is to cut it down and when the new growth starts spray it with Roundup. Or if you can catch it when it first starts to grow in the spring, spray it then. I only use Roundup at a last option, but with an established plant I will use it. And I also pick up any seed pods I see laying around and put them in the trash. I also would pick up any thistle seed heads and trash them as well. Wear golves for both. Good Luck. btw It will probably take a couple of years or so...
You might identify your thistles. Canadian Thistle can populate an entire field without shedding a single seed. A single plant can produce a network of rhizomes over 10 feet across in a single season. Mowing them just slows the process.
If chemical control turns out to be your last resort try Forefront (sp?). My pastures were in terrible shape, almost solid thistles in places. The spring spraying did a great job but there are gabillions of seeds out there so spraying twice a season will be necessary for some years. This product will kill broadleaves only. There are other issues with this product such as carry-over so be sure you speak with someone that really knows how to use it. Used properly you should get good results.