In short the HNL airport does NOT offer airport shuttle services (eg Super Shuttle, RedVan) like most metropolitan airports nor is there a connecting subway or train system (yet). There are, however, a number of good transportation options to choose from (and I offer my concluding suggestions at the end of this post):
Taxi - Walk to the curb outside the HNL baggage claim and hail a yellow cab. Approximate can fare to Waikiki is $50-75 depending on traffic and time of day. This is a good option for those without a smartphone, without prior transportation reservations, and those who have destinations off the beaten path.
Uber - personal door to door service. Within five minutes your ride will come to pick you up; the smartphone app will give you the approximate fare to your destination. A ride from HNL to Waikiki runs approximately $56 (per car, up to four people). Despite traffic or weather conditions the price Uber quotes is locked and does not change if there are delays. This is the advantage taking Uber over a traditional cab ride. Uber will take you to any destination across the island, whereas other services are limited to hotels in Waikiki. (*note: Uber is not allowed to pick up passengers from the HNL arrival terminals, however they partnered up with a taxi service (Uber Taxi) which unfortunately costs the same as a cab; the driver will walk to meet you at the baggage claim and walk you to their car. Uber is, however, allowed to drop off passengers at the departure terminals.)
The Bus - At just $2 a person, this is a good option for those traveling light and those who don't mind a 1.5 hour ride into Waikiki or multi hour ride to the northern and eastern parts of the island. There is only one bus stop at HNL airport: to get there take the elevators up one flight from the baggage claim area (to Departures) and walk outdoors to the center median. Look for the brown bus stop sign (see photo). To get to Waikiki take bus #19 or #20. Exact change only (there is an outdoor Starbucks on the departure level where you can buy something and get change.)
Go808express - I learned about this company talking with others and it seems the best option for those traveling with two or more people: direct door to door shuttle service, personal pick up from Hnl (no other parties are in the van), drops you off at any destination, and the fare is $25 for two people. Only caveat is that reservations are needed ahead of time and they have limited hours (4:30am to 9:30pm). 808.489.0000 5 Star rating on Yelp and known for being on time.
Roberts Hawaii - Unfortunately this is the closest thing to a typical airport shuttle that you'd find at most metropolitan airports (but a company that has been critiqued for its dirty politics and monopolization). To catch this bus walk to the center median curb outside the baggage claim and look for a green and white bus. Reservations must be made by calling from a reservation phone near the HNL baggage claim. The wait is 45 minutes or longer for a seat on the bus, the fare is $15/pp and it only goes to major Waikiki hotels. The additional downside is that the bus stops at every major hotel in Ala Moana and Waikiki so it takes time to get to your final destination.
The Bottom line:
If there are more than two people in your party, Uber or Go808express are the cheapest and fastest options. Taxis are the most expensive.
If you are staying with a friend, relative or small hotel off the beaten path, Uber or a cab is the best option.
If you are staying at a major hotel go808express is fastest.
If you are a back packer, a traveller headed to the North Shore or Windward side, or simply someone on a tight budget, The Bus is unquestionably the cheapest option that will take you to almost any destination on the island. Hawaii has one of the best bus systems in the country and while it will cost you time, you get a beautiful scenic tour of the island in return.
Last, you might consider combining a few options > taking the Roberts bus to a major hotel followed by a cab ride to your exact destination; or taking the city bus to the North Shore paired with an Uber ride to your exact home stay location, and so on.
However you get there, do enjoy the view out the window. The urban scape, sandy beaches, and suburban neighborhoods are all lovely textures to take in and appreciate. Aloha!