Spindly plant is an indication of not enough light. Are you using grow light or a window? when re potting bury the stem up to the first leaf. This should be done after the first set of true leaves appear. I have done it sooner but needs to be done very gently, and only if plants are falling over do to poor lighting. Hope this helps!
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Good advice from aqua. Your plants do not have true leaves yet, so they are not ready. Put them under lights and they'll start to grow some leaves. Good luck!

I have 2 24" fluorescent grow tubes (that said "full spectrum" on the package and cost 10.00 each!) suspended 12" above the seedlings. However, I didn't turn the light on until the first sprouts poked through dirt (about 4 days after planting). Maybe that was a mistake?

They look like they were grown in the dark. Drop your light down to about 2 inches above the seedlings. Al

Thanks for the helpful comments. As a neophyte, I've apparently way underestimated the light requirements. I now have my light right down nearly touching them. Is it possible to give them too much light, i.e., could leaving it on all the time counteract the spindly syndrome?
Thanks again.

No! don't leave them on 24/7. Plants need a sleep time (so to speak) 14 to 16 hours on is enough. Don't let the leaves touch the lights. They do burn if the bulbs are very hot. Another thing! Always water from the bottom, You don't want Damping off to get those tall seedlings. Good luck.