Have a patchy yard, mostly weeds that I want to turn into a cottage garden. My current method is removing (uprooting) all the weeds/grass that I don't want by hand/spade, it's a bit overwhelming as the stuff I don't want thrives and comes back very quickly. I really don't want to use chemicals. Is there a better/more efficient way to accomplish this?
The best organic practices for suppressing weeds are:
Add 4-6 inches of dry leaves on the top soil to cover the weeds. It'll be most effective if you remove the weeds before mulching. Other ways of mulching are sheet mulching, wood chip mulching. Mulching is the best practice one can follow, with or without weeds. It will keep soil moisture, improve soil structure, favors mycorrhizal fungi (plant friendly) and eventually decompose and add organic matter to your soil.
Cover crops (they are weeds themselves)
Cover crops will add nitrogen to the soil, hold soil from erosion and prevent weeds by competitive inhibition. They add nitrogen to the soil by the help of nitrogen fixing bacteria. Just make sure to chop n drop the cover crops before flowering to recycle the nutrients back to the soil.
Also there are lots of edible leguminous cover crops you can plant to have a double benefit of nitrogen fixation and consumption.
Weed mats are the best option for quick weed removal. They block the sunlight right away and kill the weeds in a few days. Go for it if you want 100% success.
Pull the weeds, till the ground, let it dry out in the sun for a bit, and then mulch. Or, till the ground, let it dry out in the sun for a bit, pick up the loose weeds, and mulch.
If the weeds have disturbed roots and the sun is strong and shining, it should make them wither and often stunt or kill those that would otherwise regrow their roots into the ground.
Anyway, tilling is a pretty effective way to get rid of weeds for a while. (If you mulch soon afterward that should work, but even if you don't mulch, it should reduce the workload.) Till goes against the no-till gardening method that a lot of people like, these days, but I personally like it. Tilling is more effecient than just pulling weeds, because you disturb all the plants at once, basically (but if the weeds are severe, you'll probably have to pull some first).
Pulling weeds is a decent amount of work, but it's probably the safest, most reliable method of dealing with weeds organically.
Covering the ground with black plastic is another option.
If you mulch with straw or hay, make sure it's weed-free. Some of it can introduce a lot of weed seeds.
The cover crops that Sudeesh Subramanian mentioned are a good option. I might add that perennial cover crops could be great to investigate. Outcompeting weeds can be effective, and you can improve your soil with cover crops, too. I'd like to use red and/or white clover, myself (you can basically make a lawn entirely out of clover, and probably still plant stuff in it).
Something that no one else has mentioned, and something I've done to establish many many gardens is sheet mulching with cardboard or contractors paper. If you put cardboard down and mulch on top, the cardboard forms an impervious layer to most weeds and they will be smothered to death before the cardboard decomposes. It's really easy! Just make sure all of your edges and gaps in the cardboard are overlapped by a few inches and you'll be fine. You can do this at the time of planting, but it is easier if you can do it a few months earlier because then you can just pull back mulch and cut a hole in the cardboard and plant into the soil. If you're doing it at the time of planting I usually remove the grass from the planting area, put the plant in the ground and then lay cardboard and mulch down over weedy/grassy areas.