My 5.75 year old was/is alot like your 3 year old. It's been interesting having an introvert as the child of a very clearly extroverted person (me. DH is more middle of the road). I've had to really advocate for Daniel in certain situations, because he is more thoughtful and, well, introverted. He takes longer to warm up in new situations, and he's cautious around strangers (and honestly - why is this a bad thing??).
When we first moved out here, we joined MOMS club so that I could meet people. We went to a playgroup once a week, and Daniel (being a very late talker) very often played alone, or did the side by side play that was very typical for that age (he was 13 months when we moved out here). As we got closer to 3, most of the others in the group were transitioning to preschool, and definitely at 4. So, we did more one on one playing with a friend from LLL, and another friend that I had made here in town. But, it was all gentle, casual stuff. And it involved decent parent involvement to make sure kids were playing kindly, and modelling good behavior.
We started going to a homeschool playgroup last year (Daniel was just over 4) - he wasn't the youngest there, since there were younger siblings of the "school aged" children there. He has really blossomed, and looks forward to playing with those kids. He is willing to play where he can't see me, and has gotten bolder in these groups because he feels safe with the kids here (and the moms). This is a FANTASTIC homeschool group - everyone is included, everyone parents gently, and the kids rarely exclude anyone who wants to play. We meet at parks and turn the kids loose, basically. THere is always a playground/sand area, and some green space for looser, "free" play. Daniel has a stutter in situations where he feels nervous or anxious, and no one has EVER made fun of him. I am SO glad I found this group.
I say all of this to just reinforce what you know in your gut - when your son is ready to venture further from the safety of homebase, he will. And, just like they tell us in all those AP books, there will be some back and forth for a little while (where he explores on his own, and then comes back to the safety of mom, and then goes out again), but he will grow.
You're raising a boy, not programming a robot
It's okay that he doesn't function just like all the other machines 
Warmly,
Amey
When we first moved out here, we joined MOMS club so that I could meet people. We went to a playgroup once a week, and Daniel (being a very late talker) very often played alone, or did the side by side play that was very typical for that age (he was 13 months when we moved out here). As we got closer to 3, most of the others in the group were transitioning to preschool, and definitely at 4. So, we did more one on one playing with a friend from LLL, and another friend that I had made here in town. But, it was all gentle, casual stuff. And it involved decent parent involvement to make sure kids were playing kindly, and modelling good behavior.
We started going to a homeschool playgroup last year (Daniel was just over 4) - he wasn't the youngest there, since there were younger siblings of the "school aged" children there. He has really blossomed, and looks forward to playing with those kids. He is willing to play where he can't see me, and has gotten bolder in these groups because he feels safe with the kids here (and the moms). This is a FANTASTIC homeschool group - everyone is included, everyone parents gently, and the kids rarely exclude anyone who wants to play. We meet at parks and turn the kids loose, basically. THere is always a playground/sand area, and some green space for looser, "free" play. Daniel has a stutter in situations where he feels nervous or anxious, and no one has EVER made fun of him. I am SO glad I found this group.
I say all of this to just reinforce what you know in your gut - when your son is ready to venture further from the safety of homebase, he will. And, just like they tell us in all those AP books, there will be some back and forth for a little while (where he explores on his own, and then comes back to the safety of mom, and then goes out again), but he will grow.
You're raising a boy, not programming a robot
It's okay that he doesn't function just like all the other machines Warmly,
Amey
