JUST DO IT. I suggest you start with easy to sprout seeds, like lettuce, onions and tomatoes.
Don't start your tomatoes NOW, unless you live in zone 6 or higher.
Make sure that you water from below, so put holes in yogurt or sour cream containers and put a bowl--plastic or styrofoam from restaurant leftovers work well--with water in it. If you saved plastic pots from plants you've bought you can clean them with a bleach solution and reuse. Don't worry about any chips or rips--I use mine until they fall apart. =b
Save bigger plastic containers for transplanting your seedlings. MANY people start their tomatoes and transplant up to the top set of leaves 3-4 times before they go into the garden. When they're really little you
carefully handle the leaves to transplant. Transplanting loses some sprouts, but the ones that survive are hardier for it. I have had little success starting beets indoors.

COOL weather starts can go out in the garden in early spring with not too much hardening off.
WARM weather starts shouldn't go into the garden until it's REALLY warm enough.
When you move out the cool weather starts you can start seeds for tomatoes and peppers in their spots. Try putting a sweet pepper in a pot that has started to rot in your refridgerator. I did that once and got about 50 plants! You just need to separate them while they are small to prevent the roots from almost locking together.
Even though my last average frost is April 15th, I don't put my tomatoes in the ground until Mother's Day. I move them onto my screened in, east facing porch for several weeks so that the breezes harden them off, but they won't get the coldest night temperatures. After that I have to harden to direct sunlight.
This Fall try bringing IN a few herbs that you grew outside. I bought chives (for pennies, on clearance) that I had out in of my beds. The original pot was about 3 in. diameter, but when I dug it out of my raised bed the root system was 10 inches across. I have it in a north facing window, happily growing. I also had a huge root system on the thyme that I'd bought, and then potted for winter inside growth. I have 13 geraniums growing next to my upstairs windows now. I took them inside in October and I'll wait until late May to put them out again. Upstairs it's dry enough to keep from soaking them--they like it dry--but when I kept THEM in my basement windows I got yellow leaves from overwatering. THIS can happen to your sprouts if it's too damp inside, like in my basement.
The mild winter made some spring bulbs put up leaves in December. I brought several inside. They won't grow again in 2012, but they'll flower in 2013 for me.
I love gardening bc when you lose a plant it's not the same as losing a pet and you gain experience.