If you're getting a jump on the season by starting your seeds indoors, soon you will find it necessary to upgrade the container you are currently using.  And, you might need to do this more than once depending on the growth of the plant, particularly tomato plants.

I start my seeds in a 1.5" to 2" cell tray.  Once the plant has put on it's first set of true leaves, I move them into larger 4" pots.  However, this will only sustain the plant so long before the roots start to curl around the bottom  of the pot and become root bound.

Roots need room to spread out and grow.  Tomato plants establish quite a thriving root system if given the proper condition.

I will only need to upgrade to a larger pot once more before I'm ready to plant outdoors in a permanent spot in the garden.

Preparing to transplant

First, collect the materials you will need.  Plants to be transplanted, Compost or potting mix, larger container, clippers, and water.

Start by using your clippers to remove some(not all) of the lower sets of leaves along the stem.
Tomatoes are great in the sense that the more of the stem you plant, the more roots it will put out from the stem.  This isn't true for all plants, so don't expect extra root growth out of you peppers. 

Prepare your larger pot by placing about an inch of soil in the bottom of it.

 

Take your plant and place your hand over the top of the container with the stem in between your fingers. 

Gently turn the plant upside down and let the dirt rest on your hand.  Lightly squeeze the side of the container to loosen the dirt from the sides of the pot.  It should pop out fairly easily.

Place the plant inside the larger container in the center.

Fill up the dirt around the sides of the old plant leaving a little room for watering.  You'll notice that the soil comes up the stem an inch or two higher than it did in the other pot.  This is what you want.

With no more pressure then you would put on your own eyeball, tap down the dirt around the plant with your fingers.  Water thoroughly.

Voila!  You've transplanted your Tomato.

What will happen over time is the little hairs that you see along the stem, when buried, will turn into more roots.  More roots means the plant pulls up more water, oxygen, and nutrients to the plant.  This translates into a sturdier, more healthy plant, that can support more Tomatoes.

This same process can also be used when transplanting outdoors.  Feel free to clip off some lower leaves and plant yet another couple of inches lower into the ground.  In not time at all, the Tomato plant will send out new roots and keep on growing. 
It might seem like a lot of work, especially if you have a lot of tomato plants, but when you harvest that first tomato and give it a taste, you'll realize it was worth it.