Of all things that you really should pay attention - is what kind of seeds you're going to use in your garden. Good quality seeds always yield good results. Now I'm not saying that you must buy the most expensive seeds in the world, but do pay attention to what kind of seeds you're getting and how they were treated. I don't buy chemically enhanced seeds or hybrids. For my garden I use only organic seeds and I keep it organic - no chemicals of any kind are on my veggies! Now I'm sure some will say "there is nothing wrong with hybrids" and I'm sure you're right. They'll grow and produce but the problem will arise after you try to gather up seeds for the next year - you don't really know what will grow from them if anything at all. For that matter I prefer heirloom varieties of seeds and organic. If you're not sure if you want to spend time dealing with seeds and all then you don't really have do that - you can just visit your local nursery and buy ready to grow seedlings and transplant them and have an instant garden. It's not really a bad way to go - you get to see what you buy ahead of time and it's already growing so it saves you time. Seeds however can save you money if you're doing something on a larger scale. For example a packet of mixed organic heirloom tomato seeds can cost upto $4. Each seedling in the store will be at least twice as much. If you grow them yourself you definitely get more for your buck. But if you only need couple of tomato plants and couple of cucumbers etc then by all means head out to the nearest greenhouse.
Now for my garden I started on my seeds in middle of February. I used small start up cups - fully biodegradable of course. Filled them with seed starting mix, soak them well and plant seeds. Don't use regular garden soil for this because it can contain fungus, weeds and can get compacted easily crushing gentle roots of the plants. How do you know when to plant? Count at least 8 weeks backwards from the last frost date in your area. That way by the time air and soil outside are warm enough to transplant your seedlings should be ready as well. It's best to transplant once your soil reaches at least 50 degrees to avoid "shocking" the new plants. Once your seedlings have 2-3 true leaves transplant them into bigger containers with regular potting soil. Because the seed mix you used before has very little nutrient it really needs to be replaced when the seeds start to grow. You need to keep them in a very sunny and warm area. If you have an overhead light that can be kept on for 14 hrs a day then it would really help your seeds. Otherwise place them on a south facing window and keep them warm and moist.
As you can see from this picture I have a mix of small cups and larger containers on a tray. I used regular baking trays to keep seeds in so I can place water in the tray and let them soak it up instead of watering individual cups - it's just so easy to damage fragile seedlings when they're only started to grow. Just make sure they don't "sit" in water all the time or they'll rot. This way I grew my own tomatoes, cucumbers, peppers, watermelon starter and beans. You can actually see one bean "waking up" in the picture. Both peppers and tomatoes will need far more time than cucumbers and beans. And both need to be very warm - 75 or 80 is best for them. Peppers especially don't tolerate cold snaps so keep them very warm and in sunny location. If you're growing cucumber seedlings indoor don't grow them too far in advance as they don't like to be transplanted when they're mature. They grow fast and will need to be transplanted after 3-4 real leaves are out. Now just make sure you feed them regularly, keep moist and watch them grow. I don't use artificial feeders but you can get easy mix in the store if you like or make a compost "tea" to feed your garden.