How do I socialize my puppy with other pets and people?

Just exposure early and often is the best way. I don't have a formula. Every dog is different. But it's just exposure. Start with a leash so you have them under control. You don't know how those other animals are going to react. So do it in a controlled setting where they're both leashed, but start small. 
If one or both of the dogs are either acting aggressively or perhaps the other way, being very fearful, do it very matter of factly. Just take them for a walk and let them get close to each other, but keep on walking. And then as they get comfortable and they realize it's no big deal, maybe then you make another pass and they get a little closer. Or maybe you stop and talk with the owner. Try to mimic a normal day-to-day situation that you might run into.
Start early. Do it often. If you can implement any of this, trust me, you will be happier for it, and you're going to have a much better, well-rounded dog when it comes right down to it.

Socializing is vital to your puppy's overall lifelong wellbeing. If your puppy isn’t socialized until they're older, you could have many problems, whether with people or other animals. So you want to start whenever they're young. If you have animals in your home, socialize them like crazy with each other. If you have family members and friends that have otherwise healthy and vaccinated animals, invite them over. Or if you feel like you can take your puppy to their home and you’re not going to have any problems as far as safety, then do that as well. But once you start getting into taking them to the dog park or daycare, we want to make sure they're fully vaccinated before you do that.