I'm using ProMix (vegetable) potting soil to start my seeds...not a soiless mix (thanks for the recommendation, Dillard!). I start my seeds in either paper or styrofoam cups (raid the McDonalds or Burger King garbage cans if you're brave.<grin> Or, maybe at "the office". Anyhow, I put 2-3 seeds in each cup of ProMix, wet it good, put the cups in a shallow plastic box and cover with plastic. When I see several of the seedlings breaking the ground I remove the top and make sure the plants are a couple of inches beneath a grow-light. I let the potting soil dry out a good bit, judging by weight, before I water. But, I water good when I water. Under my lights (two twin-tube 48" long florescent fixtures) and in the environment that they're in I water every 2-3 days. My seed-starting shelf is at my workplace so I have to be careful about weekends and insure they're watered well before leaving work on Friday.
Your tiny seedlings should only have required a slightly moist soil...any saturated soil I would imagine would have drowned the tiny roots. Also, I don't think little seedlings like those can handle direct, hot sunlight. Were the dead seedlings "crispy" or "limpy" to the touch?
And yeah, those peat cups...toss'em.
Plastic pots are reusable...I always save the ones that come with annual flowers...they're really good for starting seeds in.
But, mostly I use the paper (coffee cups) or styrofoam cups...usually around the 16oz size. For the number of tomatoes that I am planting (usually 12-14, it seems) the cups I use work well. I don't like to "pot up" to a larger pot (I'm lazy) so the 16oz cups will hold the seedlings pretty well until it's time to move them to the garden.
I'm no garden expert, but I stayed at a Holiday Inn Express many years ago....