In many cases, they can be ready when you are. Lots of folks enjoy baby beets and carrots. They are very sweet. If you want your root crops to be "full-grown" you will notice that the top of the veggie kind of pops out of the ground- from that you can judge the size of the root underneath. For potatoes, they can be dug when the blooms finish- for early/new potatoes. Or you can wait for the tops to die and harvest then. No set rules, I would say. What root crops did you plant?
Onions, beets and carrots. I've planted them all before, but this is the first time they've shown promise. Hopefully, I'm getting better at this gardening thing. :/
I usually just push some of the dirt away to see how big the bulb is before I decide. If it looks good, I pull a few at a time. Not all of them grow at the same rate so there are always a few runts. Onions I wait until I know the bulbs are large, and its been dry outside for at least 5 days. You want to pull those dry so you can cure them.
Old Guy, as you are thinning them they grow. Generally eat what you thin. Each time you thin them you have an idea of their size and maturity. Then, if you want to save seeds, save off at least 2 of the best plants for that. Then you'll know their overmature sizes.
Onions - You can use them as green onions after they develop a stem, say maybe 6 long. You can eat anything above the roots. The older they are the stronger they should get.
If any start to flower, (called bolting) you should use those then. They will not bulb up very well or store well and a hard coarse stem will grow down into them, which I dont use.
For any that dont bolt and you want to store, let the leaves start to wilt down or start dying. Then pull them and let them dry before storing them.
Carrots I just feel around the tops until they feel like they have grown fairly big around, then pull a few to see what they look like. I dont have anything really great to tell you when they are ready. Carrots will wait a little while but when the weather turns warm they can get bitter. I find it best to keep checking them and not wait too long.
Beets- I really like beets. They need to cook a long time and they can really stain things with that red color, but I really like them. And my wife really likes the greens. Anyway, when the bulbs get maybe 1-1/2 to 2 around you can harvest them. Not all bulb up the same or at the same time so you need to play it a bit by ear. Beets will poke up out of the ground a little bit so you can see how big they are. Beets will wait a while so you dont have to be in a huge hurry to get them.