Summary
- Money is essential in Tears of the Kingdom to purchase uncommon ingredients, rare materials, and arrows.
- General stores and traveling merchants offer a variety of merchandise in Tears of the Kingdom, and players should explore their options.
- Specific items like arrows, armor, accessories, and ingredients such as Fresh Milk, critters, and Hylian Rice are valuable and worth purchasing or farming for in Tears of the Kingdom.
Money is not an easy thing to come by in the beginning of Tears of the Kingdom. Players are going to want to make sure they have Rupees available so that they can restock on uncommon ingredients, rare materials, and of course, arrows.
Each general store that players will find in Tears of the Kingdom have their own merchandise. There are also traveling merchants near stables, and it won't hurt for Link to see what they have for sale. A good majority of materials/ingredients can be farmed just about anywhere, but there are a few exceptions.
Updated on October 17, 2023 by Renri Seong: Rupees are the obvious currency in Tears of the Kingdom, but there are also Poes, Zonaite, and Crystallized Charges. Link's going to need to stock up on these other "currencies" to buy the more "uncommon" items in TotK, such as Energy Wells or Ancient Blades. It's likely that most players won't have much Zonaite, Poes, etc until they're at the halfway point of their playthrough, so they'll have some time to build up a nest egg until then. The following guide has been updated to include five items that are worth buying in Tears of the Kingdom.
1 Arrows
It goes without saying why arrows are important. Players can easily burn through ten arrows in one fight - especially if they're fighting minibosses like the Gleeoks. Arrows are also the only way to use the Vow of Riju, and without it, players won't be able to call down lightning to strike their targets.
The good news is that arrows are much easier to come by in Tears of the Kingdom than they were in Breath of the Wild. Players can find at least one or two by breaking crates and barrels. However, a quicker way of getting arrows is to buy them. In Tears of the Kingdom, there's at least one merchant in each major town that sells arrows. They can be bought in bundles of five, or they can be purchased individually. Arrows also restock fairly quickly, but if players need them sooner, Beedle sells arrows albeit at a somewhat higher price than the ones found in general stores.
2 Armor (ex: Snowquill Set, Flamebreaker Set)
While more than half of the armor in TotK is found in caves or in The Depths, there are some exceptions. Armor that protects against cold weather or flames can be bought from armor shops. In fact, TotK makes it a requirement to have weather or flame protection for the Wind Temple, Gerudo Desert, and Fire Temple. Thus, players will want to save up on their Rupees and buy them whenever they're ready to take on one of the "Regional Phenomena" quests.
Usually, players will only be able to afford one piece of armor at a time. If they want, they can purchase all armor pieces to create a Set. By having a Great Fairy upgrade an Armor Set to level 2, players can use its Set Bonus. For example, upgrading and then wearing all pieces of the Snowquill Set will make LInk unfreezable, making it the ideal armor to wear when farming dragon parts from Naydra or battling Ice Gleeoks.
3 Amber Earrings (or Diamond Circlet)
In Gerudo Town, players can buy earrings. These function as "head armor," and unfortunately, they're not part of a Set Bonus. However, depending on the gems in the earrings, they can provide weather resistance, shock resistance, or increase swimming speed. Then there are the Amber Earrings and 
Diamond Circlet, which, when upgraded to level 4, have a max defense of 28. This makes them one of the best armor pieces in TotK. The Diamond Circlet may be preferred only for its Attack Up effect it grants Link.
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Amber Earrings are 400 Rupees (Diamond Circlet: 2400 Rupees), so they're not going to come cheap, but they're worth the purchase. They're also quite costly in terms of upgrades, so players will want to hold off on selling any Amber Ore. Although they're not part of a Set, Amber Earrings can be equipped with other high-defense armor. One recommended combination of fully upgraded armor is: 
Amber Earrings (28), 
Champion's Leathers (32), and 
Trousers of the Wild (28). This combo carries Link's defense to 88, which is higher than level 4 armor sets, and even stronger than the Ancient Hero's Aspect.
4 Fresh Milk
Fresh 
Milk is difficult to come by in Tears of the Kingdom, and it's unfortunately a valuable ingredient for several recipes. For one, the only general store that sells Fresh Milk is the East Wind General Store in Hateno Village. Link can also trade 3 acorns to Dantz at Hateno's Pasture for 1 Fresh Milk. Finally, he can buy 5 Fresh Milk from a traveling merchant who walks along the road between Wetland Stable and Kakariko Village. However, acorns are a necessary upgrade material for the Armor of the Wild, and the traveling merchant is easy to miss, meaning it's better to just buy the 3 Fresh Milk from East Wind whenever it restocks.
Fresh Milk is used for several recipes that are used for upgrading horses. This means they're worth saving up until players have the necessary ingredients for Malanya. This aside, Fresh Milk is also used for the Sunny Veggie Porridge recipe, which is a meal that can cure the effects of gloom.
5 Critters (ie: Butterflies, Darners, Frogs)
Players will stumble upon many critters in their TotK journey. These can be used to cook elixirs, but most of them are used for armor upgrading and unlocking Set Bonuses. Unfortunately, critters will run away the second they spot Link, making them difficult to farm.
Beedle is the only merchant who sells critters. Some critters, like the Butterflies (ie: Smotherwing Butterflies), are sold for 10 Rupees a piece, but others like Tireless Frogs are priced at 100 Rupees a frog. While the pay may be high, buying critters from Beedle is something to keep in mind if players are struggling to farm them on their own.
6 Hylian Rice
For some reason, 
Hylian Rice isn't too common in Tears of the Kingdom. They could be farmed by cutting down grass in West Necluda in BotW, but they're much rarer in TotK. Fortunately, the grass near Lurelin Village yields Hylian Rice, but Lurelin Village is far from the "starting area" of TotK, so it may be a while before players can set foot on Lurelin Village's shores.
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Like 
Fresh Milk, Hylian Rice is used in a few of the recipes for horse upgrades. Players are also going to need 20 Hylian Rice for the "Lurelin Village Restoration Project" side quest, which is highly recommended. Those struggling to farm Hylian Rice in the wild can buy it at most of the general stores for a fairly cheap price: Slippery Falcon (Rito Village), East Wind (Hateno Vilalge), Gerudo Town, Goron Gusto Shop (Goron City).
7 Goron Spice
Goron Spice is used in curry recipes such as Vegetable Curry and Herb Saute. It's also only sold in Goron City. The cost is 16 Rupees, so it shouldn't make much of a dent in the wallet. On the downside, the general store only stocks 3 at a time, and unlike the Fresh Milk, there isn't a side quest to trade items for a jar of Goron Spice.
8 Cane Sugar
Cane Sugar's another ingredient that's quite common in general stores. However, it's another ingredient that can be farmed in the wild. Cane Sugar costs 12 Rupees, and there's usually 2-3 bundles depending on the shop. It can be found at the following stores: Great Deku Tree's Navel (Korok Forest), Slippery Falcon (Rito Village), and Goron Gusto Shop (Goron City).
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Cane Sugar can't be cooked alone. It needs to be cooked with another (compatible) ingredient. Recipes that use Cane Sugar will generate desserts, which aren't as filling as hearty meals like Meat-Stuffed Pumpkins, but they're good to eat if Link doesn't need to recover much HP.
9 Goat Butter
Goat Butter is another common ingredient as it's sold in the general stores at Hateno Village and Kakariko Village. Like Cane Sugar, it's priced at 12 Rupees, and stores usually stock up to 5 bars of Goat Butter. The reason why players will want to buy Goat Butter is due to the amount of recipes that call for this particular ingredient.
Big meals like Salmon Meunire and Prime Meat Stew need at least one bar of Goat Butter. If players intend on stocking up on meals that restore a lot of hearts, they're going to want to make the purchase. Those that intend on upgrading their horses will need Goat Butter for dessert recipes (ie: 
Apple Pie and 
Carrot Cake).
10 Bomb Flower
The Fuse Ability in Tears of the Kingdom was the perfect excuse to do away with Fire/Ice/Shock Arrows and Bomb Arrows. While it saves on rupees, some fans weren't pleased with the replacements. Bomb Arrows in particular would have been very useful in TotK thanks to all the cavern walls that need to be destroyed. Instead of the Sheikah Remote Bombs and Bomb Arrows, TotK has a returning item from the Zelda franchise: Bomb Flowers.
Link can find Bomb Flowers in caves, but he can also buy them from certain vendors. A Gerudo Merchant in Gerudo Town will sell 1 Bomb Arrow for 40 Rupees; she stocks a total of three. The Bargainer Statues in The Depths and the one at Lookout Landing will give Link a 
Bomb Flower for 16 Poes. While it's better for TotK players to get Bomb Flowers from caves, the Gerudo Merchant and the Poe Statues are a useful resource if Link has some Rupees or Poes to spare.