Sometimes gardeners shy from using plants with a propensity to seed themselves around the garden, for fear that these self-seeders will outcompete neighboring plants or upset the overall planting design. These are valid concerns, yet gardening with self-seeding plants has its upsides:
It can be of great benefit to bees, butterflies and other pollinating insects. Self-seeding plants typically flower abundantly, providing plenty of food for these insects while also creating a lively, colorful display.
It requires less in the way of resources. If plants are happy to "volunteer" in your garden, they are happy with its conditions—no need for supplemental water, soil amendments or fertilizer.
It offers surprises that can be gorgeous, and it's the chance to turn "garden design" into a partnership with nature, rather than a struggle for control.
Creativity is not lost. Although plants will pop up where you didn't plan for them, you don't have to leave them there. You can, but there's always the option to pull or move them.
Getting started with self-seeders:
Experiment with those that may already be in your garden. Allow some seedlings to develop wherever they pop up. If you decide you don't like how they look later, you can still pull them or move them. Ask local gardening friends for chance seedlings that appear in their garden, and for their observations of plants that self-sow and to what degree.
To encourage volunteer seedlings, don't mulch. The mulch that suppresses weed seeds will do the same to desirable seeds. You will need to weed, but the amount of weeding could taper off after a few seasons, especially as you encourage a thick tapestry of plants that are ideal for the site. If you can't skip mulch altogether, opt for a shallow layer of a fine material such as shredded leaves or pine needles, and/or wait until summer to apply mulch, so that volunteers have a chance to come up in the spring.
It goes without saying, but avoid including invasive plants in your garden. The goal is to encourage some self-seeding, not overtake a neighboring ecosystem. Research the plants you're adding to your garden and consult with local extension agencies, botanic gardens and wildflower societies for planting ideas and locally native species.
Use straight species or naturally occuring varieties rather than hybrids or cultivars, which often have been rendered sterile to prolong their bloom time and therefore will not produce seed, or if they are fertile, their offspring may be weak or show less desirable traits. If you use non-sterile cultivars, be aware that the seedlings may not precisely resemble the parent, particularly if you are growing more than one cultivar, as they may cross-pollinate. (This isn't necessarily a bad thing.) For more on this topic, see this post from South Dakota State University Extension.
Thin your volunteer seedlings so that they can develop. Sometimes a bunch will pop up in one place; when they begin to crowd against each other, pull some and compost or transplant them.
Familiarize yourself with how your plants look as seedlings vs. how common weeds in your area look, so you won't mistakenly pull a garden plant and leave a weed. This knowledge will accumulate over time; when you're starting out, don't be afraid to let a potential weed develop a bit or even flower if you're unsure. (Just don't let it go to seed if you realize it is a weed!)