How do I make a lot of money?

Fortunately there are a few options to make money without even grinding too much for it. Some of those options are available from the very beginning while others will be unlocked in later stages of the game. Here are some methods ordered by the time in the game you can unlock them.

High-Profit Investigations

Unlocked by: Completing quests/Killing small monsters
Availability: As soon as investigations are unlocked (start of the game)
Profit: Medium-High
Time required: Low-Medium
Passive or active: Active
Pro: Available from the very start of the game, you gain extra materials per quest completed.
Con: Actively requires you to do a special quest, investigations don't always show up (RNG)

How it works: Keep an eye out for investigations that have an increased zenny reward (marked with an exclamation mark next to the reward). Often those quests require you to slay 5-15 small monsters which can be done in under 5 minutes. You can also use one of your log-in vouchers on these quests to double the amount of Zenny gained.

Cultivating Might Seeds

Unlocked by: Completing optional Botanical quests up until "Persistent Pests"
Availability: Mid Low-Rank
Profit: Low-Medium
Time required: Medium
Passive or active: Passive
Pro: Early availability, requires one time setup and is passive after that
Con: Only mediocre Zenny rewards, takes up a cultivation slot

How it works: Might seeds have been a popular money making method in previous titles of the series and remain so in World. The method is simple: You cultivate might seeds that sell for 140 zenny each and you complete quests normally. Using fertilizer will further increase your profit.

Bandit Mantle

Unlocked by: Completing the optional quest "Redefining the Power Couple"
Availability: Late Low-Rank
Profit: High-Very high
Time required: None
Passive or active: Passive
Pro: High income, passive method, income increases with monster difficulty
Con: You sacrifice a tool-slot (Defense, Utility)

How it works: By far the most efficient money-making method in MH:World is the Bandit mantle. You can equip the tool on a hunt and it causes your attacks against a monster to sometimes drop a trade-in item as a shiny drop. These items can then be sold for anywhere between 1000 and 5000 zenny per piece. At the start of high rank you can expect somewhere around 10000 zenny extra per quest if you use the mantle and this amount increases the further you progress into the game. A great thing is also that it's completely passive if you are hunting big monsters (which is what you should be doing most of the time).

The Argosy

Unlocked by: Reaching High-Rank
Availability: High-Rank
Profit: High
Time required: Low
Passive or active: Passive
Pro: High income, passive method
Con: Costs a decent chunk of resource points

How it works: The moment you reach high-rank the Argosy will be available. It's a ship you can send out to gather materials which you can then buy for resource points. Here you can choose the option to buy trade-in items. Every 3 quests you complete the Argosy returns and you will be able to buy whatever you've ordered. This is essentially trading excess resource-points for Zenny.

As a general tip regarding trade-in items: If you sort your item-box (R3 on PS4) they will always show up on the last page and are labelled with "Trade-In". Those items can be sold without much thought and are only in the game to fill your pockets.