Seedlings grown in cell trays need a sufficient root system in place before transplanting outdoors. Waiting until the seedling develops its second true leaf is one way to determine if a seedling is ready or not - but that is not always a hard and fast rule.
Having a strong root system is the best indication. The three best ways to achieve a healthy root system are to use a bottom watering system, plant in proper trays that encourage downward root growth, and with the use of a proper fertilization regimen. We generally don't like to give direct advice about fertilizing as it is different for every individual and situation, but buying a well rounded potting soil will help to ensure that your seedlings are covered nutritionally.
(Check out: The Easiest Way to Water Your Seedlings)
The length of time a seedling can stay in a cell tray depends on the size of the cells. Deeper and larger cells both allow for extended growth as the plant has more room. Seedlings otherwise can become root-bound if not given adequate space for the roots. Typically, after sowing the seeds, the cell trays are used for around 3-4 weeks before transplanting occurs - whether it be to an outdoor plot or into a larger container.
One thing to keep in mind with transplants, is to choose your sizing according to how long the transplant must stay indoors. This will factor in on the cell size needed for the seed type. Here's a great reference guide for sizing cell trays.
(Check out: Selecting the Right Cell Tray)
Farmers have moved into using cell trays in order to increase the amount of transplants that can be grown in a smaller space. Many have their own tips and tricks to finessing a seedling plug out of a cell. Bootstrap Farmer cell trays are specially designed for growers by growers to make this chore a bit easier. Here's a great video highlighting how to remove the plugs from the cell trays.
When growing in seed cell trays, the seedling can start to get thin stemmed from lack of light. If the seedlings are not getting direct light, this often happens. The best way to handle this is to either transplant the seedings into a new pot or hardening them offand transplant them outside. Either way you will want to cover them with soil up to the lowest set of leaves in order to help strengthen the seedling.
Sowing seeds into cell trays and planting starts into the ground offer some major advantages to the grower. Planting into cell trays helps increase the amount of starts that can be grown in a smaller amount of space. This is a huge advantage for growers who are growing for quantity. Another advantage of growing out seedlings to transplant is a higher seed success rate. You only transplant strong seedlings, therefore generally are able to have more growing success.