Let There Be Light: For seedlings to grow properly, they need light. And lots of it. Chances are that you don't have enough natural light to grow healthy robust seedlings, even if you have a south facing window. If seedlings don't get enough light they will be spindly and won't make it to healthy adulthood. Don't be alarmed though, setting up an artificial light system can be easy and not expensive.
I have a simple set up in my basement using inexpensive metal shelves. I have attached shop lights using "S" hooks and the chains they came with, so they can be raised as the plants grow. Sometimes if you're growing several types of plants under one light, one side of your shop light will have to be higher than the other, as plants grow at very different rates. I have fitted the shop lights with one cool and one warm fluorescent bulb. I also plug these lights into inexpensive timers, so I don't have to keep track of turning them on and off. Once the seeds have germinated I set the lights to go on for 14 hours during the day. I also keep them as close as possible to the seedlings - two to three inches at most. It's amazing how much better and faster seedlings grow if the lights are that close.
Use Self-Watering Seed Starting Systems: I will never start a seed in a small peat pot again. They just dry out too fast. Self-watering seed starting systems are becoming increasingly popular, and there are many to choose from. I've tried several and my hands down favorite is the APS Seed Starting System from Gardeners Supply. I'm also very partial to my own design of a free, self-watering seed starter that you can make from a supermarket pie plate and some string.
Use a Good Seed Starting Medium: One might think that given all the seeds that grown in the ground and do just fine, that you could grow your indoor seeds in garden soil. Bad idea. Seedlings are very susceptible to a dreaded fungus called “damping off.” You know you have it when all your seedlings are fine one minute and the next day, they have keeled over, dead as a doornail. With all the work starting seeds requires, it makes sense to give seedlings the best chance for survival by using a sterile planting mix.
Feed Your Seedlings: Most sterile planting mixes don't have don’t have any built-in nutrients at all. For awhile seedlings get all nutrition they need from the seed itself, but when you start seeing leaves you’ll want to feed your seedlings with a diluted solution of liquid fertilizer.
Ventilation and Wind: Seeds are really meant to be planted outside in the elements. They are designed to get sun, rain and wind. By planting indoors you are really attempting to fool Mother Nature. Lots of us are planting our seeds in our basements, the part of the house that is probably least like the natural world, with air that doesn’t move.
Not surprisingly, it turns out that moving air is an important factor in helping seedlings to develop a robust root system and strong stems. To approximate wind, try setting a fan on low near your seedlings. If you do, you will get sturdier plants. I also gently run my hands along the tops of my seedlings for a couple of minutes a day to give them even more of a workout.
Read the Seed Packet: Most seed packets have a wealth of information. First, they will tell you how deep to plant your seeds - a critical piece of information. Most packets will tell you if it's even ok to plant the seeds you are considering indoors - some plant just flat out don't like to be transplanted and are better off started in the container garden in which they will live. The packet will tell you how long it should take for the seed to germinate and how many weeks before the last frost you should start your seeds. Keep the seed packet for the life of the plants. Chances are there will be information that you will need at some point (particularly if you've thrown the packet away).