Organic Vegetable Benefits

Just don’t use chemicals. End of story.

You are asking a huge question. There are big books written about this subject. Since you ask a broad question I will answer it broadly.

Growing plants organically is a lot more than avoiding chemicals. It really is about healthy soil.

http://www.rodaleinst.org is one of the most complete sites devoted to organic growing

But start with the soil. Fertilize with compost, green manures, natural mulches. Start a compost pile ASAP

Have a crop rotation plan (this is important for even the smallest gardens)

Find certified organic seed to plant

Control insects through a variety of methods including crop rotation, exclusion, hand picking, beneficial insects, selective spraying (most organic pesticides, and yes they do indeed exists, are broad spectrum so you want to use them sparingly so you do not kill off the beneficial and neutral bugs that make up 97% of the bug population), trap cropping etc..

Hoeing and mowing to control weeds