As a boy, most summers were spent visiting my grandparents where I often marveled over their lush garden.  Sometimes we'd visit in the spring before the garden even got started.  My grandpa, who was kind enough to put up with my numerous questions, took me into his very small greenhouse where he had rows of pots with tiny plants growing in them.  He explained that starting some plants early was necessary because of the risk of unexpected freezes in spring.

Later in life when I stared my own garden, I never forgot what my grandpa taught me.
Spring is unpredictable.  One minute it can be 75 degrees outside, and the next day a cold front can fall out of the north, blanketing us in snow, ice, or just freezing cold temperatures.  Therefore, starting seeds indoors or in a greenhouse is a good idea, if not absolutely necessary.  

I've seen an ever increasing number of Seed Starting mixes on the market.  Supposedly, these provide a sterile environment in which your plants can get started.  
There are also those lovely discs you pour water over and watch the compressed peat expand in the tiny mesh sacks.
Or, those cardboard looking pots that you can plant into the ground and eventually break down.

I've tried them all.  

The mix, when transplanting, all falls apart.  Even wet it just falls away from the roots and the plant. I just don't like that.  But the larger thing for me is that the little seedling doesn't have any food in a sterile mix.  I'm a busy guy, so I'll admittedly leave plants in their pots a little too long before transplanting into larger pots.  I personally like to have a little more substance to my starting mix.

With the peat discs and pots, I've just not been very impressed with the way the roots are able to penetrate beyond the containers.  Bad root structure = less productive plant.

What should you use?  That's really up to you and you style of gardening.  I can only suggest what has worked for me over and over again.
Every serious gardener I know will agree with me that Compost is “Black Gold.” Good compost can easily be made at home.  Some cities will offer loads of compost made from the collection of organic material from grocery stores and neighborhood yard waste.  Some will even offer it in convenient bags.  

I use a mixture of compost, peat moss, and some perilite.  The compost is nutrient rich, the peathelps retain moisture, and the perlite provides aeration to help promote root growth.

This helps my seeds by providing a nice environment for them to germinate and the plants to set their first few sets of leaves.  

Seed starting pots or trays are readily available at most stores with a garden center. Just get your pots, soil, and seeds together and get planting.
Follow the directions on the back of the seed packet.  Some go down 1” below the soil, others are best on top with a very light covering.
Water carefully, but all the way.  Keep moist, but not soaking.  Place them in a nice, warm spot.  The top of a refrigerator is a good place.

After a few days, you'll start to see the seedlings emerge. 

Keep under grow lights or in a greenhouse.  You might need to transplant into larger pots depending on your time frame.

When the risk of frost is gone in your area, transplant outdoors and watch your garden grow.