I usually start my tomatoes about 8 weeks before the last frost, peppers 8-10 weeks and cucumbers and zucchini about 4 weeks (they grow fast). Not sure when the day of your last frost is? You can always check here.
There are many things you can start your seeds in, but this is what I normally use. Plastic cups work well also, just make sure you poke some wholes in the bottom for drainage. You will also need some kind of seed starting mix. You can usually get a good sized bag for around $5-$6. Moisten the seed starting mix with some water, but don't get it too wet to the point you have mud. This will give the seeds a good start to their germination. Your seed packet should tell you how deep to sow your seeds.
When starting seedlings indoors you really need some type of artificial light. If your seedlings don't get sufficient light they can get "leggy" and weak. This is the grow light system that I personally own: Hydrofarm Jumpstart Grow Light. This is great for the beginner that isn't growing a large amount of plants. This year year we are building a larger grow light system which I will share in another post.
Once the seedlings have started to sprout you want to take the cover off. Too much humidity can cause fungus and bacteria. At this point you want to make sure you keep the light about 2 - 4 inches away from the plants. Because plants indoors aren't getting natural sun, you will want to make sure you plants are getting about 16 hours of light. A lot of people put their lights on timers so they don't have to remember to turn them on and off all the time.
My tomato seedlings should be coming up any day now, soon hopefully followed by the peppers and onions.