What were common crew ranks on US Navy flying boats and bombers?

Any one from the Aviation Branch could, depending on the needs of the service, be assigned as a crewman in a USN aircraft (though some more or less likely than others), and some not of the Aviation Branch, as well. Typically the usual ratings were: Chief Aviation Pilot (CAP); Aviation Pilot (AP1c, 2c); Aviation Chief Machinist's Mate (ACMM); Aviation Machinist's Mate (AMM1c, 2c, 3c); Aviation Chief Electrician's Mate (ACEM); Aviation Electrician's Mate (AEM1c, 2c, 3c); Aviation Chief Radioman (ACRM); Aviation Radioman (ARM1c, 2c, 3c); Aviation Chief Radio Technician (ACRT); Aviation Radio Technician (ART1c, 2c, 3c); Aviation Chief Metalsmith (ACM); Aviation Metalsmith (AM1c, 2c, 3c); Aviation Chief Ordnanceman (ACOM); Aviation Ordnanceman (AOM1c, 2c, 3c); Chief Parachute Rigger (CPR); Parachute Rigger (PR1c, 2c, 3c); Chief Aerographer's Mate (CAerM); Aerographer's Mate (AerM1c, 2c, 3c); Chief Photographer's Mate (CPhoM); Photographer's Mate, 1st Cl. (PhoM1c, 2c, 3c); Chief Radioman (CRM); Radioman, First Class (RM1c, 2c 3c); and, not particularly unusually, Seaman 1c and 2c.

Insignia on for sleeve ratings marks can be found here

Before April 1942 when all Naval Aviation Pilots (enlisted pilots) still in enlisted status were redesignated in the above Aviation Pilot ratings, one could find NAPs from ratings outside the Aviation Branch ratings; ones rate was not what mattered, what mattered was finishing the NAP course at NAS Pensacola. So one could easily find, for example, torpedomen, gunners mates, yeomen, photographers, and even the occasional Seaman 1c as a rated pilot in a navy airplane.

Enlisted Navy Job Classifications (1945) may be of interest. See pages starting at 125.

Most pilots, and copilots in multi engine aircraft, were officers, anything from a warrant officer (though most of those were commissioned as lieutenants (jg) or lieutenants by the end of 1942) up to commander in a typical operating squadron, though as noted above, there were enlisted pilots. Most enlisted pilots were also offered commissions and most accepted though there were some who remained enlisted for the duration . . . they were not required to accept a commission.

The USN operated just about every multi engine aircraft as the USAAF, with the exception of the B-29, and some that the USAAF did not. Here’s a quick and dirty list, though it mistakenly lists the P2B, the Navy designation for the B-29 as the USN only had 4 of them and none were acquired before April 1947 . . . beware of the internet or at least Wiki.

Crew duties were as the typical USAAF crews although it was not unusual to find an enlisted bombardier, usually of the AOM variety rating. Navigators could be either officers or enlisted. Gunners on planes so equipped, once the training pipeline kicked had to go to gunners school, and while one’s rating was immaterial these slots were predominated by the ARM, AMM, and AOM ratings. Before such places as the Naval Air Gunners School at NAS Hollywood, Fla, were established gunner training was at the squadron level. For crew served aircraft, crews, pilots and all, were generally trained together in an advanced training squadron as a crew and were eventually assigned to a squadron as a complete crew, either as a replacement crew or as a crew in a squadron working up.

Some information on the training command from The Navy's Air War may be of interest. The entire book may downloaded here

And, obviously, not everyone flew combat. In fact, the largest squadrons, measured by number of pilots assigned were in the Naval Air Transport Service; for example in March 1945, the transport squadron VR-11 had 581 officer pilots assigned and 160 officer navigators, ranks ranging from commander down to ensign.

Later . . . now that I've slept . . . I neglected to point you also in this direction (it's terrible, I have all these documents in PDF and then I have to figure out from whence they came). This is Introduction to Naval Aviation published by the Office of Aviation Training, OCNO, in 1946, and provides considerable background of historical interest. Chapter X covers training. https://www.google.com/search?q=Introduction+to+Naval+Aviation&rlz=1C1GCEA_enUS892US892&source=lnms&tbm=bks&sa=X&ved=2ahUKEwjS4qSetMXrAhWfoHIEHebuDecQ_AUoAXoECA4QCQ&biw=1346&bih=909