My experience (not anywhere near Manitoba), is that if you mow regularly, tall plants can't thrive and short ones do. We got a lawn of mostly grass, with some clover and other weeds. But, it was green, soft, walkable, and kept the ticks and horseflies away.

Mowing is probably more successful faster if you start in the spring, as your yard wakes up.

My experience with Round-up (which is the advice you will hear from others) is that it doesn't work for deep-rooted plants, they recommend not planting for 4 months and it may take repeat applications.

You can also try to smother the weeds, but that also takes time and some stubborn weeds can survive for a long time.

The final option is to rototill several times. You want to mow first, so you get rid of any seeds and you may chop up some roots into little bits that will continue to grow, but then, you'll mow them down when you have sod.

Meadows are lovely to look at, but depending on the climate not pleasant to walk in: ticks carry diseases aside from being very unpleasant. Anything you mow regularly will eventually be all grass. If you really want to grass the cheapest thing to do is to hire someone with appropriate equipment to mow your meadow once. Then get out there with a lawn mower every week during the growing season. You'll have passable grass by late spring. In the early fall (right after first frost) over seed with a grass suited to your climate. Grass loves cool wet weather, and in the spring there is a lot of competition from other annuals.

All that being said, in your shoes I would mow paths. Then you have place to walk in your lovely meadow. Meadows are much more interesting as well as more natural than sod, and they don't require constant mowing, a chore I'm going to bet you aren't going to like once you have extended experience with it. Many many people go to great lengths to create a meadow for exactly these reasons.