Like the others I have a primary care doctor. They are part of a family medicine office, which is essentially a GP surgery. They do children as well as adults, but I actually take my baby to a pediatrician for primary care. The office is just better geared to kids. It's also common to see an OBGYN at a Women's Health office for all gynacological and obstetrical care as it's easier to go direct rather than through primary care which then needs a referral for surgery, procedures etc.
To add some more detail on referrals, whether it's needed often depends on your insurance. I have a PPO insurance plan. They are slightly more expensive, but you can generally go straight to a specialist without a referral and you have access to a wider range of doctors. IN some cases you need a referral- e.g. fertility treatment, psychiatric care.
Some people have HMO plans. These are generally cheaper but you need to go to Primary Care first and get a referral, and often there are fewer doctors who take the insurance.
Re: recommendations, IME they've always recommended good doctors but I'm sure there are informal networks between doctors that mean they are more likely to recommend some over others. Recently my OBGYN asked me who I was seeing for another issue and whether I liked them as she wanted to have more options to suggest for other patients.
Re: nurses, my experience is that they are still very pivotal to the running of hospitals, although I feel like doctors have more authority. When I gave birth recently I mainly saw nurses except for when I was pushing (my OB delivered) and my OB and the pediatrician doing rounds. However, there were nurse practitioners from pediatrics and the NICU who would be called on. That's my only experience of US hospital care though.