One of the most important things is light. You need to have about 16 hours of light for begonias. I know it seems strange since they are a "shade" plant in the garden, but under indoor lights they need more light than many other seedlings. Also, I have found that they fertilizing them very early with diluted fish fertilizer will help too. I dilute the fertilizer in a misting spray bottle.
side note: when I used to work at the wholesale growers, they did run into issues for lack of light. It is really important that if you are starting them under lights that it is at least 16 hours per day.