Having trouble finding an activity or playgroup that you and your preschooler might enjoy? Try forming a play co-op with other parents of preschoolers in your area. When you have the kids to your home, you'll have a mess to clean up but depending on how many parents and children you include, you'll then have two or three or more playdates that involve no clean-up at all.
If you don't send your child to preschool, he will depend solely on you to make sure he has a social life (or indeed any kind of life outside the home). As your child's exclusive caregiver, you are entirely responsible for what he learns, does, and enjoys. You become the sole source of new experiences, field trips, playdates, and introducing new things and people to play with.
If you choose to keep your preschooler at home, you'll need to find alternative ways to give him the opportunity and the encouragement he needs to make friends. (You'd also be well advised to find ways that allow your three-year-old to spend at least a short time without you or his special caregiver so that he gets used to the care and company of other adults.)
Especially if your three-year-old has never experienced daycare, he will probably find it easier to make friends in small groups rather than in large ones. So try to find a playgroup, class, or activity that might both further your child's social contacts and give him the opportunity to experience small group activities.
If your child and her playmate are doing something unsafe or tempers flare, merely issuing a warning ("Be careful!" or "Calm down!") just doesn't cut it. Take a hands-on approach in enforcing such safety rules as "one at a time" on the swings or "no pushing" on the jungle gym. And whenever possible, separate and distract children before anger turns to aggression or violence.
Besides finding group situations for your child, try to arrange for playdates and special outings with one or two other preschoolers. Encourage your child to feel free to ask friends over to your home (or to ask you to call their parents to invite them over).
When you're hosting a playdate, keep in mind that three- and four-year-old playmates still need at least some adult supervision. Tempers flare much too quickly among preschoolers to leave them alone safely for more than a few minutes at a time. If you're the responsible adult in charge, you have a duty to make sure all children in your care remain safe from themselves and from one another.
Fortunately, this doesn't mean you need to watch the children like a mother hen. Indeed, you may find that you can get more accomplished and have more time to yourself when your child has a friend over to play with than when you are her sole source of entertainment and companionship. But you do need to stay close enough to be able to step in whenever anything threatens your child's—or your guest's—safety.