I don't use the method described in green below anymore. I have developed a much better way to start seedlings.

I'm busy getting this year's gardens going so the photos will have to wait for time being. In the mean time, here's the process:

Use the plastic egg cartons from Sprouts Grocery.

Put a tablespoon of water in each pocket.

Add 2 tablespoons of fine sand to each pocket.

Add another tablespoon of water to each pocket.

The sand should be moist not wet. 

Create a spreadsheet like this with a set for each egg carton.

Put a number on the upper left corner of each carton.

Put as many seeds as you like in each.

Cover the seeds with a thin layer of sand.

Close the carton and place it in a warm sunny window or under grow lights.

Check the moisture level at least a couple of times per day. 

When the seedlings get to about a half inch tall, they are ready to move to your regular starting packs with potting soil.

For most seeds, scoop out the whole pocket and drop them an inch or so onto a flat container like a large lid.

The sand will fall off and you can pick out individual seedlings to plant in your starter containers.

Some seeds like tomatoes, squash or melons may already have a root a couple on inches long at this point.

I try to create a hole deep enough to drop the root into while keeping it straight.

Some seedlings like foxglove or coral bells will be thinner than a hair. I put the whole cluster in to the starter without dropping it.

Seed sprouting time can vary from a couple of days to 2 or 3 weeks. 

Why I use this method:

It is fast, easy and reliable.

Requires far less space to start.

Sometimes I only have a very small number of seeds  of a given plant. I want every  one to have it's best chance.

Some of my seeds date back as far as 1997 and may not be viable any more. (Most still germinate.)

The roots don't get tangled or damage as in the paper towel method below.

The seeds don't get mixed up.

Easy to accommodate different germination rates.

It's much easier to maintain the correct moisture level.

LABELS

Copy all the seedling names into one column.

Sort them.

Set the font size to 20 pts. and make it bold.

Set the print area.

Set the paper type to transparency.

(If the printout comes out thin or rubs off, set your paper type to Thick Paper or Card Stock instead of Transparency.

Printout on a laserjet  printer (Inkjet will not work) on transparency film.

Cut into strips the width of the page using a paper cutter.

I put the label in the starter cups and then transfer it to the garden when I plant the seedling. 

You can print multiple pages or duplicate individuals as needed. 

Fold them in half when you put them in the ground so they won't fall out or blow away.

A table knife works well for pushing them into the ground.

These labels will easily last the whole season. 

Old Method I Used to Use.

Plants, like peppers and tomatoes, can benefit from a longer growing season.  This is accomplished by starting them indoors a few months before the latest frost.  Our typical latest frost in Albuquerque is April 15th.  Some years we may have to wait a couple weeks longer.
It does take some time and effort to do this, so I don't do it for plants that don't need it. If you're only planning on planting a few plants, it's a lot easier to just buy them in the store.  Last year I planted ~200 peppers and over a dozen tomatoes.  Some of the plants I just buy.  Most of the ones that I start early are ones that I can't find in the store or get as early as I want.  Last year the green chile plants I bought started out well and where ahead of my plants.  In the long run, my plants where healthier and produced much better.  By selecting my seeds over the years, I have been able to produce superior fruits.  It's not sure-fire, so I plant seeds from different years.
I store my seeds in small sealed zip-lock bags in side another zip-lock in the freezer.  I have been able to keep pepper seeds viable for over 5 years.  Make sure the seeds are completely dry before you seal them or the will mold.  Another reason for starting your own seedlings is to avoid genetically modified plants. I'm not going to get into that here.

Other plants that you may want to start early are okra, artichokes(globe), brussel sprouts and some flowers.  I do not recommend starting squash or melons for transplanting.  They don't like transplanting and it is difficult to do without damaging the vines when you move them. It is better to use a cold frame for them, if you have to.