If you live in a cool area, you can use a greenhouse or cold frame.
Otherwise get the most from your growing season and start seeds indoors... window sill, top of the fridge or the coffee table.
Or think about an enclosed porch, a garage, a shed or basement.
Growing seeds inside is pretty straightforward. Give them air, light, water, warmth — and they'll do what they're meant to do — sprout.
The best way to water seed pots is to put them into larger trays and fill with water. This way the seed pots soak up water from the bottom. Otherwise spray carefully with a fine rosed watering can.
For ongoing water, pour more water into bottom trays or use a mister spray bottle.
Have labels and marking pen, and use them immediately you have sowed each variety of seeds. Too bad if you forget, because it's guesswork otherwise.
For a good seed-raising mix, see: Vegetable Seedlings
So unless you have brilliant natural light, try and arrange lights over your seedlings. Keep this indoor lighting just above to your seedlings... about 7cms (2-3 inches) away, moving up as they grow.
Preferably use fluorescent or one of the latest energy saving lights. They are more economical, they cover a larger space, they run cool (without burning the plants) and they use less electricity than incandescent lights. Purpose built grow lights or fluorescent grow bulbs are excellent.
You may need to have these lights on for at least 15 hours a day to provide optimum conditions for seedlings... they MUST have good light and for a long time.
Heating the soil will speed up germination, even though the surrounding air temperature may be chilly. Using seed heating pads or mats, even hot-water bottles and electric blankets under your seed trays, can make it possible to produce soil temperatures in the 80's.
The difference between soil temperatures of 15-21°C (60-70F) and 26-30°C (80-85F) mean seeds germinating in two weeks or one week or less — that's half the time!
Mini bench glasshouses can be bought, or made, eg: Bend some coathanger wire or similar to form arches to go over your seed pots or trays, then cover with clear plastic or plastic bags.
Clear plastic supermarket trays with hinged lids that are used for cakes, salads and soft fruits like berries etc, make nice little mini glasshouses. If you don't buy them, then like me, you'll see them in other people's homes or their recycle rubbish.
The less root disturbance the better for seedlings, so this double handling method is not recommended unless you are careful and skilful.
If you sow your seeds in individual pots or in trays with plenty of space between each seedling, you can simply keep your seedlings there and transplant straight into their permanent spot in the garden when large enough.
Start toughening them up, by putting them in a protected part of the garden and bring them back in at night. As they get stronger, they will be well prepared to withstand the actual conditions full time in the garden once you turf them out for good.
Starting seedlings indoors gets you in the gardening zone much sooner in the year and revs up the old spring fever! And to keep you in the mood, have a read of these other helpful guides...
Vegetable Seedlings: How and why to grow your own vegetable seeds
Seed Varieties & Products: Types of Vegetable Seeds
Use Your Own Seeds: How to Save and Store Vegetable Seeds
Vegetable Sowing Guides: Seed Sowing guides for different climate regions
Germination Guide: Times for Seeds to Sprout, Stratification, Scarification & Chitting of Seeds