Here are some things to look out for:
Don't over water the seeds
It's tempting to keep the soil constantly wet because you think that plants need lots of water to live. They don't need that much.
To check if they need to be watered, leave one pot/container/space in the same area but without putting any seeds in and stick your finger in there all the way to the bottom. If the bottom is dry, THEN water. (Water this empty container as much as you water the seeded ones. After awhilee, you won't need to do this.). A lot of people mistake dry top soil for dry bottom soil. BAD MISTAKE. Overwatering will often cause root rot and invite pests.
Space them out as best as you can
Although some people prefer plucking out tiny seedlings to make space for others, I think that's the lazy way out and wasteful. Just spend a bit more time with individual seeds as best as you can, spacing them out when planting
Know when to transplant
The earliest you should transplant is when the seedling has a decently-sized pair of main leaves. All seedlings have their first leaves, which usually don't look anything like their actual, main leaves. Only when a seedling has their first pair of main leaves should you try transplanting, not earlier. I mean you could, and I've done it, but it usually stunts their growth for a month or so while it recovers from shock at such a young age.
In the photo below, you can see that most of these seedlings have two initial leaves and a first main leaf. Can you spot them?
Try to keep the rootball intact when transplanting
This means not letting a lot of soil right under the plant stem fall away. If the roots become too loose, the plant will need to spend even more time establishing a 'foothold' in its new soil after planting. You'll just be delaying its growth and flowering.
It seems seeds love to be planted on or right after a new moon
Don't ask why...