What is the optimal way to make Money?

Edit 2: Updated again March 2018 to include new methods introduced in the Beyond patch, as well as remove information that was invalidated by that patch.

Edit: This is an updated answer as of Feb 2018. I knew passenger missions were the fastest and most lucrative missions, but I put it to the test and tried to find out what the upper limits were. I reinforce the same points here as my original answer but give (I think) some clearer examples and comparisons, plus a step-by-step explanation of my current money farming plan.

The absolute fastest method of making money that I have found is by doing passenger missions, specifically transport missions to stations that are far away from the arrival point. The step-by-step is listed below, and further down I'll list quick comparisons with the other lines of work mentioned.

Long-Distance Transport Missions

Find a station that you can dock at, has a passenger lounge, and is really far away from the entry point of its system.

The easiest way to do this is to go to the eddb.io station list and search for a station:

Set Min Landing Pad filter to the smallest size landing pad your ship can use.
Leave the Max Distance to Arrival field blank
(Optional) Only set Include Planetary to "Yes" if you own the Horizons DLC
Set the Reference System to your current location
Click Find Stations

This is searching all stations in the database that you can land at. After the search completes, click the Arrival Distance column header, and that will sort the stations by their distance from the system arrival point. Sort this column so the largest distances are at the top.

For example, if you were in Sol, this would be your returned results:

long-range-stations

So Katzenstein Dock in the 36 Ophiuchi system is 4,218,205 LS from the arrival point (that's about an hour of supercruise, FTR). Transport missions become more valuable based upon the length of your supercruise trip, so any missions going to Katzenstein are going to be very lucrative.

Note: The recent Beyond update has drastically reduced some of the longer-distance rewards. There no longer seems to be much of a difference between a 500,000LS trip and a 3,000,000LS trip, so Katzenstein is now a bit overkill considering the travel time involved.

Go to a system nearby that destination, within ~20 LY or so. Find a station with a passenger lounge and look for any missions that are going to Katzenstein Dock. In my experience, after a local faction is friendly with you, they will offer 4-10 million credit missions for low-tier requests (5 Economy passengers, 2 First Class), and 10-15 million credit missions for high-tier requests (32 Economy passengers, 18 First Class, 6 Luxury). When allied, factions will give 10mil / 15mil missions that are basically identical, except one is Tycoon and one is Elite rank.

It may require switching between Solo and Open Play a few times to get the missions to pop, but it's possible to (modestly) do a single run with an Anaconda, Beluga, or a Type 9 that nets you over 30-40 million credits.

It just requires:

Getting to the system that's near the system containing the long-distance station
The time to find the missions you want to stack up
The travel time to the station (4 million LS = ~1 hour of supercruise, 1 million LS = ~30 minutes)

Why passenger missions?

If you can jump ~20ly, most transport missions can be completed in one jump, sometimes 2 or 3.
Avoiding criminal / wanted passengers eliminates 90% of the dangerous missions; avoiding secretive passengers will avoid a lot of failures just for being scanned.
Even if VIPs (or others) make demands mid-trip, you can deny those demands and maybe just lose a little passenger satisfaction, a possible bonus, or it can have no consequence at all.
Missions often stack, meaning you will often find multiple groups of passengers going to the same location
Single missions can often net over 1Mil credits, and I've seen Luxury passenger missions pay out 4Mil creds for a one-jump trip
Visiting the same stations over and over will raise your rep with local factions, giving you access to the higher tier missions as well as just more missions in general, and thus higher payouts per jump

I personally have about ~36 Economy, ~24 First class, and ~8 Luxury seats in my Beluga, and I'm able to pick up multiple missions at nearly every stop that are going to the same location; on average, I make anywhere from 800K to 2.5M per jump by stacking missions. This equates to me sitting down to play for about an hour or so, and walking away about 5-15Mil creds richer, depending on my luck.

Mass Transport Missions

The recent Beyond update has introduced larger transport missions, which sometimes require 50+ seats for a certain class of passenger. This can be accomplished with 2 size-6 Economy cabins, but takes quite a few cabins for First Class. The rewards for these trips, depending on the distance, can be 3-8 million credits in my experience.

Wing Missions

Wing missions are a new addition as of the Beyond update and offer some very lucrative rewards, but they also allow you to share the workload. I find these to be the 2nd most lucrative money-making method as of the update.

Any wing mission you accept can be shared with your wing members, and each person contributes to the overall mission goal. If someone messes up, a partial reward can be accepted if certain criteria are met. The only wing members that are rewarded are the people who are in your wing when the missions is turned in, so make sure to wait for your buddies who helped to be online (and in your wing, with the wing mission accepted) before you accept your cash, or they will be out of luck. Everyone gets to pick one of the three offered rewards, and everyone gets the full reward (no profit sharing) regardless of their contribution (yes, even with zero contribution).

Massacre missions usually want you to kill a lot of people (100-200) for a few million bucks (biggest I've seen was 10mil / 190 kills). Definitely easier with a wing.
Delivery missions are point-to-point haulage missions (300-3000+ units), meaning the cargo you transport is given to you at the mission port, and dropped off at the destination port. The cargo is unique, so if you are delivering cothing (haulage) and you blow up, you will never get the full reward, even if you go find and buy more clothing (which is cargo, not haulage).
Source and Return missions are a little different from delivery missions. You are not given the cargo to transport, you have to go find it, "acquire" it, and bring it back to the mission port. The profits are usually larger than the delivery missions, but that's because you have to (usually) buy the cargo. Case in point: the first Wing mission I tried was $27mil credits to source and return something like 3000 units of gold. I was excited about the giant reward until I realized I'd be spending about $21mil just to acquire the gold I needed to return.

Mining

The upside to mining is that besides fuel, limpets, and initial equipment purchases, mining is essentially all profit. With multiple lasers (5) and multiple collection limpets (8) on a Type 9, I used to rake in about 2Mil credits worth of Painite / Platinum / Palladium from Pristine Metallic rings for a 1-hour mining session. It's a grind, but its peaceful and I found it fun. That being said, I can make 2Mil a jump sometimes with passenger missions.

Basic Missions

I have personally found that basic mission-board missions don't have great payouts, and require a lot of work for the money.

The notable exception: SRV missions. There are Sentry Skimmer missions ("Destroy X number of sentries near Base Y"), and sometimes Power Grid missions ("Shut down the power grid at XYZ!") that have multi-million credit payouts (1-3 million in my experience), but the sheer time it takes to do them makes them less viable than passenger missions.

Trading

Personally, I find this to be the most grindy, annoying way of making money. Basically, you need to ensure that each jump you're making from station to station is netting you a greater profit than you are spending in terms of time and fuel. Each station is limited in what it sells, and those goods are only desirable to certain types of economies, so to really be efficient at trading, you will need to rely upon 3rd-party tools that will plan out a route for you, "guaranteeing" a profitable trip each time.

Say you buy 100 units of X at 1000 credits below galactic average, and sell them for 1000 creds above average; if my math is correct, you just made 200K in profit on that trip. While that's nothing to sneeze at, I don't even pick up passengers offering less than 500K anymore, if that gives you some perspective on per-jump profitability.

Planetary Landings

I'm not really sure what profitability there is in just random planetary landings; there are always things to collect on planetary surfaces, from materials to cargo to occupied escape pods, but they aren't inherently valuable, and don't seem plentiful enough to fill up a cargo hold very fast. It's also annoying (in my opinion) to fill up your tiny SRV cargo bay and have to call in your mothership to unload every time. It's very fun to explore planets, you can find some interesting and beautiful sights, but aside from Mission Board missions that request you to specifically do something on a planetary surface, I don't believe there is a straightforward way to make a lot of profit from random landings.

Exploration

If you're geared up for it and don't mind being lonely, people do say this is profitable. Generally your scanners, scoop, and FSD are the only real requirements, and your quality of life will be enhanced with a big fuel scoop. But, the payouts depend on where you go, and if you get back safely.

The goal is to travel far, scan lots of interesting things, and get back safely to a station. That can be a very dangerous and complicated thing, or a nice peaceful jaunt into the black, depending on your luck. Here is a great infographic found on Reddit that shows profitability of different sources:

explore targets