Beets are a great crop to grow, when they mature you actually get a twofer, the roots are the main crop and additionally the leaves are wonderful steamed. Don't plant the beet seeds too closely, the roots need room to expand as they mature. The beets will actually mature at the soil line so when the reach the size you want simply pull them out of the ground. I've never had good luck with the golden beets but they are beautiful when they do grow.
Carrots require deeper soil so if your bed is the standard depth build a small mesa of soil to give the carrot roots room to grow deep. Plant the seeds as directed on the pack and keep the soil moist to aid germination. You can cover the carrot seedbed with newsprint to hold moisture, check frequently and when you seed the first sprouts remove the newsprint. As you watch the carrot tops grow, you'll see the shoulders of the carrots right at or just below the soil line. Use a finger to scratch the soil around the stem to look for the top of the carrot.
Potatoes are generally easy to grow and you can find the seed potatoes in the garden centers now. Dig a trench about 6" deep, space your seed 10-12" apart and cover the seed potatoes about 8" deep. Once the potatoes are about 1' tall you'll mound soil around the plant halfway up the stem covering the lower leaves. As the new potatoes swell underground this additional soil will prevent the new potatoes from swelling out of the ground and getting sunburned. New potatoes are ready in about 60 days, full maturity in 90 days.