First, prepare the soil.
Seeds should not be planted in weedy soil; they will not be able to compete. You have two choices here.
The traditional method of soil preparation is spading or forking by hand. You want to loosen the soil enough to let in air and make the surface crumbly, remove weeds (or bury them) and incorporate amendments to enrich your planting area. Start at one end of your bed and work a shovel at a time from one end to the other. Then break up the clods with a rake or fork, and level the bed. You want a fine texture so seeds don't fall into cracks too deep for them to find the sun. All amendments should be spread evenly and incorporated into the top 4-6” of soil.
Tilling is often used to prepare a seedbed and remove weeds. It is a quick and easy way to get a workable seedbed. However, it will make the soil dry out faster, so beware if you are in a drought zone. It also damages soil structure and the diversity of soil life. If you have a choice, choose a plow or spader over a rototiller. If you are using a rototiller, use a shallow setting and go slowly as possible. Never till perennial weeds that spread by underground runners, like Bermuda grass, quack grass, bindweed, or sheep sorrel. Every piece will root and make a new plant. Pull them or shade them out with cardboard first.
In the past decade, soil science has drastically changed our understanding of the role of soil organisms in building fertile soil that both encourages root growth and provides nutrients. The best, most fertile soils are not tilled or disturbed and are always covered with plants and/or mulch. No one has done more to teach the labor-saving and soil-building possibilities of no-dig gardening than Charles Dowding, a market gardener in the UK. His website, books, and YouTube channel have easy and inviting explanation of no-dig gardening for the home gardener. Recently, a wave of young vegetable farmers have transformed market garden techniques by focusing on encouraging soil life. Their methods are well-documented in the No-Till Grower's podcast and YouTube channel, as well as books by Jean-Martin Fortier, Jesse Frost, Brian O'Hara, and others.
Amendments are minerals, or plant & animal residues that are added to make the soil more fertile or to correct imbalances. If you are digging your garden, mix them into the top layer or soil. In No-Dig gardens, they are scattered on the ground and covered with compost. Worms and other organisms will work them in for you.
The best way to know what amendments to add is by getting a soil test. Another way is to ask local gardeners, farmers market vendors, or your cooperative extension service. The best all-round soil amendment is compost. It cannot burn or damage plants, unlike more concentrated fertilizers. Compost helps your soil retain the water, oxygen, and carbon to feed soil microorganisms and plant roots, as well as containing enzymes and plant hormones for quick seed germination and healthy root growth.
If you have acid soil (more likely in areas with rainfall over 40 inches a year,) you may want to add a small amount of lime or wood ashes. Depending on your climate and underlying rock present, your garden may have other specific nutrient needs.
Then sow your seeds.
Now the seeds or transplants go in. Consult the label on the packet and make sure you know these things: Where to plant—full sun, part shade, or shadier. When to plant—before or after your last frost date in spring or first frost in fall. How deep to plant—if in doubt, figure on twice the seed's thickness. But some seeds need light and should be covered only lightly if at all. Others need deeper planting to access moisture and have a strong anchor. (Corn & sunflowers are typical of the latter.) Last, how far apart to plant.
Some people (usually with larger plots) use the row system: you till the entire area and put your vegetables in single rows and blocks as needed, with no permanent paths. In that case, rows should be a 1 ½ times to 2 times as far apart as the spacing of plants within the rows. (Depending on how much access you need and how you plan to weed and water.)
Many home and market gardeners use the bed system. This has beds of a standard size, with permanent paths between. Within the beds, you can plant in lengthwise rows, crosswise rows, or staggered rows (a honeycomb pattern, which fits the most plants into the space but is harder to hoe.) Some grains and cover crops are sown broadcast, where you scatter the seed as evenly as you can and rake it in slightly, (using a short back and forth motion rather than a long sweep with the rake.)
To plant a row, use your hoe or trowel to make a furrow, by drawing the corner of the tool to make a line as deep as your intended planting depth. If you use drip irrigation, put your row next to the drip tape. Scatter the seed in the furrow, slightly closer than recommended spacing. Then use the soil from the uphill side of your furrow to cover the seeds. Water well, and keep moist until you see sprouts.
If you're using transplants
Find out the spacing they need, and get something that can roughly measure that for you as you plant. It's a good habit to know the width of your own hand with fingers extended and together--this is the easiest measure. Or get a stick and mark it. When you know the pattern you want--along drip tape, in a honeycomb pattern, or whatever, decide on your starting point and go. For small transplants from six-packs or modules, you can use a dibber to make the holes (a dibber is a sharpened dowel or branch you poke in the ground.) If you have bigger plants in post, use a trowel to lever a hole or slit for planting. Do not let the plug of soil from the pot protude from the ground--it will dry out. Deeper is better than shallow. Firm in well. In hot dry climates, use any excess soil to make a little saucer or dam on the downhill side or around the seedling to hold water.