Hi, it's one of those case by case things. Usually, for us, if we do the "inviting", the other parent brings the child to our house (as they've been "invited"). The pick up depends on what works best for that particular situation: "We're heading in that direction later anyway, so it would be no problem to drop off your child at your place later on" or "We eat dinner around 6 so would you be able to pick up your child by 5:30 or would you prefer she stay and have dinner with us?" or just simply "What time would you like to come by and pick up your child?". If the playdate is at someone else's house, unless the other parent says something like "we can swing by and pick up your child after my daughter's soccer practice", I assume I'll be taking my child to their home. If it's a new friend, I prefer to drive to and from so I can meet the parent at their house and do a little recon on what kind of people they are. And again, unless the other parent offers to drive my child home because it's convenient for them, I assume I'll pick them up after the playdate.
Absolutely set up playdates for your child, absolutely! It's the best way for them to find which children they're comfortable with. I just moved across the country and have found the playdate "etiquette" is pretty much the same here as it was in my last city. It probably is in your town, too. Since you're new to the area and don't know too many people, I think it's fine to tell the other parent "Why don't you and your child come by around 1pm. If you'd like, stay for a few minutes yourself to make sure your child is comfortable. I'd love to get to know the parents of my child's friends, so maybe we can chat for a few minutes before you leave." If your child is young, chances are the other parents are also new to drop-off playdates and may have the same concerns as you. There's nothing wrong with telling the other parent "we're new to this whole drop-off playdate thing, so please bear with me". Common sense, simple courtesy and friendliness will help you set up good playdates for your child. And don't get offended if once you have a playdate at your house, you don't immediately get invited to the other child's house for a reciprocal playdate. People are super busy these days and sometimes it is a real hassle to have a playdate at their house. It doesn't mean they don't like your child or they're being rude on purpose. My son has friends who've been to our home many, many times and he's never been to theirs. I know that both his friend's parents work fulltime, sometimes weekends, they have activities, and a dog (which my son is allergic to) and another daughter, and a very small home. Coordinating a playdate at their house is really troublesome for them. My son likes this kid, so I have no problem having him over to our home and don't expect a return invite. Hope your child has many many successful playdates and you get to know a bunch of parents!