Potatoes did terrible at the farm, and I've never grown them at home before, so I have no tips there.
For carrots, dig deep, make sure there aren't rocks or hard lumps in the soil. Weed and thin carrots ruthlessly. My farmer friend used to joke that farming is 90% weeding carrots. It's true. You need to get all of the weeds out and then out the seedlings to give a decent amount of space in between. We just used the scottslawn organic fertilizer and got a decent yield on okay-ish land. When harvesting, sort them out. Really good big ones can go in cold storage (frig or root cellar), small or damaged ones should be used soon or preserved. Don't worry if you puncture a carrot with your fork, just don't store it long term. I lived on damaged and too-small carrots for awhile. haha
Easy preservation ideas that worked for me: slice, blanch and freeze on a cookie sheet, then bag. These are great to toss in at the end of cooking soup, added to casserole, mashed and eaten with butter, candied, or added to smoothies. Slice large carrots or use tiny ones, pack into a canning jar, pour pickling solution over the carrots to fill the jar (vinegar, salt and sugar is what I use). Cap and put in the frig. These will keep a few months (this is my go-to for making a lot of different pickles; radishes are my favorite). Dehydrate slices or julienned carrots to add to soups.
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That is a very nice suggestion. Thanks.