Vaccines

It depends on how risky the area where you live is regarding the diseases they're being vaccinated for in those last two rounds. Some places you're probably fine with a little basic risk mitigation but in other places various diseases can be super endemic. It can vary year to year, too. Personally, I'd take your vet's advice as they have personal knowledge of whether there are local outbreaks or not.

Note that vaccinations aren't instantaneous, either - it takes a while after the jab for dogs to build up the antibodies necessary to prevent disease.

While I agree that it's important to minimize the possibility of exposing your puppy to disease before they're fully vaccinated, it's also important (many behaviorists say more important) to begin socializing them before the window closes. At 13 weeks, you're already getting very near, if not past, the prime socialization period for your puppy. "Socialization" isn't about taking them to the dog park & letting them run amuck around tons of dogs, or even walking them around a heavily dog trafficked park, it's slowly exposing them to novel things in a safe & controlled manner & making sure the experiences are GOOD ones. This can absolutely be done without putting the puppy at high risk for disease exposure.
The following article explains it well:
I also highly recommend you download a copy of this new (excellent) book: