You can plant directly from your cells as you suggest. You should clip off any doubles right away. In the soil of a six cell you will have only enough soil for one root. When potting up I would not be using peat, but a good quality potting mix. Al

Okay, I will make sure there is only one seedling per pot. I am still a little confused about the soil though. When moving my tomatoes into larger pots, can I still use the sterile, Jiffy Seed starting mix? Or do I need a potting soil?

Stores sell both potting soil and potting mix.
Many people just say "potting soil" when they mean the "mix"
I know of no cases when you want the soil unless you are short of common dirt. Use the potting mix. You could probably use up your starter mix for potting, but it is a little finer. And cost more. Potting MIX is bark, peat and perlit. I usually just use it for starting mix too.

So it would be alright to use up my seed starting mix to move my plants to bigger pots?

starting seeds and potting up is not brain surgery. It is just that some things work better than others. when a kid we used garden dirt and old tin cans...now that was about as bad as it can get. You can save the starter mix for net year, use it to pot with or maybe mix it in with some regular potting mix. What ever feels good to you. You will learn as you go.

I would disagree with Al on the lettuces. I do not pot them on or single them but harden them off while still in multiples in the cells and then transplant straight into the ground by gently pulling the clumps apart. I find that the fact that they are different sizes and not all perfect large individual specimens means that they will grow at different rates and I can get a long succession of lettuce from a single sowing. I deliberately keep some back in the cells and plant out at intervals. You don't necessarily want all your lettuce ready at once. But this works in my climate because we have months of cool damp weather appropriate for lettuce. You may need to plant them all at once for a shorter cropping season.