If it's really shady, more sun will help. With this year's heat wave for most of the summer, shaded gardens tended to do better.

I would say do what you can to prep your soil this year. You have options.

Cover all the weeds and grass with a thick layer of newspapers and top with mulch (grass clippings, compost, leaves, etc). Leave it until next spring. The weeds and grass should be dead. You might be able to just plant right in it, or you can till it all up in the spring to mix everything together.

Note that tilling brings weed seeds to the surface and they will germinate, so without added mulch next year, the weeds need to be tended to regularly.

Another option is to spray everything with roundup - wait 2-3 weeks and then till it up. You can amend the soil with compost too.

When you say cardboard products, you do mean a few large sheets of cardboard right? The fewer/larger the number of pieces, the more effective it will be. I would start ASAP.

I would Roundup all those weeds you are referring to, not just outside the garden area, but if you're unwilling to do that, that's understandable too.

I'm not sure where you are in ON but if it's anywhere between Ottawa and Windsor (Southern ON), do succession planting. Plant your leafy greens first in late March/early April, then harvest them and transplant the toms and cukes in late May. You're blessed with a much longer, warmer season than me in Saskatchewan :)

That weed you mentioned that seems to have underground connections needs to be dealt with. By shade, if it is only from distant buildings or trees, you might survive, but if it is from nearby trees, you have a big problem that will not let your garden produce well.

As you are learning, the crop must be thinned and spaced properly. If the soil you plant into is deep and fertile, it will need continual enrichments as you remove produce. If the soil is weak, fertilizer is definitely needed.

You could take a photo of the weed & post on the Name that plant forum to learn what you're dealing with & how to eradicate it. Digging out the root with a tool is work, but just might be the last time you pull it! I've been half heartedly pulling some dandelions in the lawn this summer to feed to the chickens and yesterday got out the weed hound, a tool that pulls the roots up. Over & done with now and worth the extra steps of carrying the tool out with me. Sometimes the smart fix is more work.

Spread farmyard manure 6-12" deep on top of the cardboard before the other compost ingredients then the worms will do the work for you! If you don't want the weed seeds to germinate, don't till the soil. There are more posts here and on the Organic Gardening forum about that method. Also read about interbay mulch on the Soil & compost forum. I've used it for new gardens and to renovate old ones filled with weeds & grass. I won't dig up sod or spray weed killer anymore because this other way is so much better & works!

Try craigslist if you don't have friends with animals. Skip pig manure, but any of the others will make an amazing improvement in your soil.

Mounded beds work and you don't need wooden frames, though the frames do help define the garden and keep the soil raised. I have both types.

Straw or spoiled hay is great for the paths and mulch. In fall I spread all I have to cover everything nicely. Then in spring it's so fluffy underneath. You can get the free hay sweepings from the feed store if you ask. They sweep up & toss at the end of the day. One store near us puts it in a separate dumpster.

Round up will kill annuals, but not all perennials.
If the plant has tubers,bulbs, fatty roots for storing food then they will come back.\If you spray & cover with cardboard the you will do better.
I am an organic gardener with 3 of the hard to kill perennials & have to dig them out, because I do not use round up. I till around my garden to keep the grass out.
Mulch will help to , this is a first year bed 12x20.
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Same bed later in spring.
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10 x 20 means you're going to have to have some paths. I like 3-4' wide beds. Some plants take up lots of space. Square Foot Gardening book will give you great advice on spacing. You can start a lasagna bed now. Hats off to jolj. I would not use roundup either. If you start with wet cardboard or a 6-8 sheet layer of wet newspapers, most weeds don't come up through. Alternate layers of whatever like grass clippings, chopped up leaves. Top with composted manure. My first bed was an industrious 24" tall, composted down over the winter to 12" and is now level with the rest of the ground.

Think about how much you are planting. There's only two of us so even a 10' row of swiss chard or beets is really too much. Figure out some succession plans. I have kale, beans and scallions growing where spinach (wintered over) and sugar snap peas grew. In your colder zone it might be trickier to get 2 crops out of the same space. I use a fabric covered tunnel to protect bush beans from a frost that usually hits our garden around Sept 18. Once past that date, we have a couple more weeks of warm weather.

I put wet newspapers covered with grass clippings in my pathways. If I change the layout next year, everything is decomposed. Get a good hand weeding tool. For me it is something similar to a cobra headed weeder. We have grass and another weed that travels by runners.

Keep in mind that if you keep the spring garden weeded, the veggie plants usually get big enough to shade out new weeds. Keep in mind that nature abhors empty soil. If you pull up grass now and leave it uncovered, you'll just get new weeds. Either put the sheet composting or lasagna bed in place or wait until spring. You can also cover a weedy bed with black plastic which holds down the weeds. A neighbor starts her news beds that way. In the spring, everything's dead and she can till. But, with a lasagna garden, you don't need a tiller.