I take Millie almost everywhere, every place I can obviously because she's in training as my service dog.
One place I don't take her is horse therapy, because I can't work with the horses and her. She's great with horses but I would think it would be inappropriate and useless to either of us to leave her somewhere and then go to the arena.
She's prettydamnreliable, but I still take things very slowly with her and try to judge what kind of stresses there will be in a place and how long we'll be there, what places we'll be afterwards, etc. My mom says I underestimate her, but I just don't want to mess up doing this.
She's gone to the grocery store, department stores, the mall, the theater, the bookstore, people's homes, doctor's appointments, therapy appointments, the insurance company, pet stores, a school, and lots more places.
Yes, I still have problems with people wanting to pet her. I usually intercept them and say "Please don't pet her, she's working and needs to focus." Sometimes when people ask I put her in a sit- or down-stay and let them. I have problems with adults and I can always count on when going into a busy store that there's at least one small child screaming to their parents about the puppy. Luckily Millie is very good and just leans into me and I take her away from there.
The only time I ask for permission is when in a private home. Sometimes I won't go or will wait until a different time if I can't take her. One exception was when visiting a friend's mom in hospice care.
I take Lucy everywhere I can, to private homes, pet stores, and other places dogs are allowed. Incidentally, I'm taking care of the animals of 4 different households and I take them along with me when I do my jobs, excluding when I'm lunging the horses.
While I'm not as adamant about petting as I understandably am with Millie, Lucy gets distracted much more easily. I try to let people pet her as much as possible when I'm not trying to do something in particular with her. She loves petting and when I take her out people stop me constantly.