What is the best way to get rupees?

Cooking is the fastest and most efficient way of making money. Using the right ingredients, you can create dishes and elixirs worth 2400 rupees, or more. See this question for examples, as well as the specific formula used to determine the value of the resulting dish/elixir. I suggest farming Moldugas/Lynels and using their guts to make elixirs.

Another way to make money is by playing some of the many minigames available. There's a gambling minigame in the beachside town in the southeastern part of the world which costs 20 rupees to play, and which will give you 100 rupees if you win (33% chance). By save scumming (saving before the minigame, playing, then reloading if you lose), you can earn money basically for free. It's a very tedious though. There's also the flying minigame, the horseback riding minigame, sand seal racing, and others which offer similar profits.

Finally, a method that I've used is to farm dragon parts. The three dragons spawn in certain spots very reliably, often several times per night. See this question for details on where to find the best place be when farming them. Try to always get the horn shards - they sell for the most money of all the possible parts, plus they are used in several quests and equipment upgrades.

If you're really strapped for cash, you can also sell your armor sets to some shops for a couple thousand rupees. However, I strongly advise against this - the various armor sets in the game offer powerful bonuses, and they are pretty much one-of-a-kind. If you ever do sell your armor, it's kind of like taking out a loan - you'll get the money you need right now, but if you ever want the armor back, you'll have to buy it again from a special NPC in Tarrey Town - and they are very expensive.

I played through the entire game, collecting gem stones (Ruby, Diamond, Sapphire etc.) as I went, and never found any particularly useful purpose for them beyond the very occasional item crafting and upgrading purposes. I kept expecting them to be useful someday, but they really weren't. Certainly no need for a huge pile of them anyway.

Sell off all but 10 (just in case) of every gem type you have and never look back.

There's a Gerudo woman in Goron City who buys gems at better than market price, if you're interested in maximizing the profits, but she only buys one type at a time.

I'm finding that pretty much everything has some use somewhere down the line. Plants and fruits seem to be the most abundant and least useful but you don't get an awful lot for them.

However the advice you are (frequently) given in game has worked the best for me; sell gemstones. Luminous Stone sells for 70 Rupees and is very easy to find. Just grab a hammer and climb a mountain at night. Midway through the game (having not sold any) I found myself with about 50. They do have various in game uses but they are reasonably easy to come by so I wouldn't worry too much.

Rubies, Sapphires and Topaz are even better. They sell for a lot (180 - 240 if I remember right) so you don't need a lot to make money. You can tell which ore deposits have gems in (they are shiny and gold) so mark them on your map and come back frequently to see if they've respawned. Or hunt Talus, once you get the knack of killing them (and particularly if you have access to the Goron special weapon) they are pretty easy and give you loads of gems and ores. The gems do also have various uses in the game, but I've found they're more useful to sell then any other potential use so far. Amber and Opal are needed for various armour upgrades but you can find quite a lot of them so they are good for selling too.

Monster parts are tempting, as some can fetch quite a lot, but you need them in abundance for upgrading armour so I only tend to sell things i have loads of (100+) and even then I save them for Kilton...

So far I've not really had to wait long for anything money related and haven't found anything where I've thought "Damn, wish I hadn't sold that!" by mostly just selling gems.