The following ingredients are must-haves to make the recipes:

All-purpose flour: Normal flour you can buy just about anywhere. Store brand is just as good (and usually cheaper). You’ll need some extra to roll out your doughs, too!
Granulated sugar: If you have a food processor or Magic Bullet, you can create smaller grains of sugar (caster in the UK) by pulsing the sugar for small intervals. This creates a smoother texture if you’re making creamy frostings or lighter bakes, or if you don’t want to buy powdered sugar.
Powdered sugar: Needed to create buttercream. A dusting of powdered sugar on cooled bakes also ups the visual appeal!
Butter/margarine: They are interchangeable! Butter hardens quicker in the refrigerator, so it’s ideal if you have the time to refrigerate/freeze after you cut your cookies before popping them in the oven. If planning to keep for longer than a few hours, margarine-based doughs are easier to work with immediately out of the fridge. Butter creates a little more body in a bake than margarine.
Eggs: I’m sure they’re much cheaper in the mainland… Most of the recipes here call for large eggs.
Salt: Table salt is fine. If you’re fancy and have different salts, you can try substituting that for slightly nuanced flavors.
Milk: I don’t like drinking 2% or higher fat milk, but they’re good to bake with. Dried milk powder (such as King Arthur brand baking milk powder) is a suitable substitute (when added with water). I’ve also been experimenting with evaporated and coconut milk, as well.
Heavy whipping (double) cream: Needed for the buttercream and custard. You can make heavy whipping cream by mixing milk and butter (it must be butter!) if in a pinch.
Baking powder: Store brands may call it “double acting,” which is fine. Be careful using too much before your bakes taste metallic.
Yeast: You can try the yeast packets first before you buy a jar of yeast. The yeast packets are good because they’re shelf-stable; if you buy the jar, it’s best to keep it in the fridge. Buy the regular yeast (usually has a dark red accent color on the label).
Vanilla extract: Usually a must have in all bakers’ homes. For most of these recipes, imitation is good enough.

These ingredients are good to have:

Baking soda/bicarbonate of sodium: If you watch British programming (e.g. Great British Bake Off [Baking Show in the US]), they call baking soda ‘bicarbonate of sodium.’ This is just another raising agent.
Spices: I’m sure you have spices, but some good basic baking spices include allspice, cardamom, cloves, nutmeg, ginger, and cinnamon. Spices are where you can kind of play around without too much consequence; you can tinker with amounts to make them stronger/weaker, but just be careful not to use too many different spices (esp. the same amount) before they all cancel each other out and you don’t really taste any particular one.
Flavorings/extracts: As long as any extract isn’t your main flavor, you can use imitation without feeling bad about it. If you decide to make anything your main flavor (e.g. vanilla, almond, lavender, etc.), splurge and get the real (pure) extract. I usually try to have orange, almond, and lemon in my cupboard at all times, and lately I’ve been into using lavender, rosewater, and rose syrup. You can also get regional spices/flavors; I have matcha and kinako (roasted soybean powder), as well as za’atar, elderflower, black currant, and crushed mint.
Dried fruit, chocolate chips, and nuts: These can transform these basic recipes into much fancier and heartier bakes in an instant. I usually always have raisins, blueberries, and cranberries, as well as mini chocolate chips. You can make a ganache using chocolate chips and heavy whipping cream in the microwave, which also transforms bakes to another level!
Sprinkles: I love nonpareils on practically anything. Haha.
Pasteurized egg whites: These are a lifesaver when you need a meringue or royal icing and you don’t want to have leftover egg yolks. The best part is they don’t run the risk of having bits of yolk, so you know your stuff will come out nicely.
Liqueurs: Flavored liqueurs are another way to get subtle flavors in bakes. I have amaretto and brandy, as well as a yuzu liqueur. Orange liqueur is on my list to buy.
Specialty flours: Cake flour is weaker, while bread flour is stronger than all purpose flour. This results in either a lighter or heavier texture based on what you use. I’ve written which flour to use in the recipes, but I’ve noticed all purpose still works if you don’t have the budget to buy other flours. A couple of the recipes require self-rising flour, which can be made at home with all purpose flour, salt, and baking powder.
Different sugars: Brown sugar is always good to have, as well as raw (in place of demerara) sugar; demerara is a little more molasses-like, so you could experiment with adding a touch of molasses with the sugar. They’re good to substitute part of your granulated sugar if you find the result a little too sickly sweet; none of these recipes should be though!
Baker’s Joy or PAM Baking Spray: These have flour in the formula, which produces less tackiness on baking goods and a slightly less sticky finish on your bakes when they come out. You can always spray regular cooking oil (or grease with butter/margarine) and dust with flour after to save money.
Sure-Jell and lemon juice: These help the jams become more firm, but you can always boil down without them. The lemon juice also acts as a tart flavoring to reduce the sweetness of jam since they contain a lot of sugar.
Ready-made jam: In Hawaii, most times I don’t have the budget to make homemade jam as often as I’d like. Thus, I like to have a selection of ready-made jam or jelly to use. Apricot is a good flavor to have as it’s hard to distinguish in small quantities, and can be used to plug up cake holes before decorating with buttercream, fondants, or other frostings. Ross and other discount stores are good places to look for different flavors you may not find in your local market.
Cocoa powder: It doesn’t add too much to the dry ingredients despite using a lot to get that cocoa flavor through. You can also mix with sugar and milk to make homemade hot chocolate!
Shortening: This creates a nice crispness that butter alone can’t create. I use this mainly in my chocolate chip cookie recipe. I prefer regular shortening over butter-flavored shortening.

The following tools are must-haves in my kitchen:

Measuring cups and spoons: American recipes tend to use cups, while most other recipes tend to use weight measurements. I have cups (for both dry and liquid) measurements just in case (or when I’m too lazy to measure exactly), while I rely on my spoons more since that’s more universal.
Kitchen scale: I couldn’t get my bakes all the same without a scale, and I’ve felt its importance throughout the past year. I bought a digital one from Walmart (for less than $20) and it works just fine, so you don’t really need to splurge on a fancy one. Make sure you have a flat, level surface on which to use it (I have a glass stove top range, so I use that).
Rolling pin: Even better if they have the guides at the ends to ensure uniform thickness. I have a faux stone rolling pin (that I also bought from Walmart), as well as the one with guides at the ends.
Cutting board: The larger, the better. It’s something I can wash quickly after I’m done rolling out my doughs. If you like baking and you think you’ll be looking at making pastry, think about getting a marble/stone one from Ross.
Baking sheets and pans: I hope Costco still has a sale on Nordic Ware baking sheets… Regardless, I have light cookie sheets and dark pans. Sets are a good place to start, and you don’t need to splurge on these yet until you know you really like baking. I have two active baking sheets, one 9 x 13” pan, a few round cake tins, a couple loaf tins, and a cupcake pan. If you want more specialty pans, I also bought a tart pan and popover tin. Try to look for a cupcake/muffin tin without grooves on the outside of the cup for easier cleaning.
Baking paper (parchment paper): These really help with keeping your pans clean and cleanup time a lot shorter. If your dough is on the drier side and your paper isn’t stained or oily, you can reuse it by leaving it in a Ziploc bag in the freezer. I usually reuse mine once or twice before throwing it away. Wax paper is good to use when you’re not baking, but I don’t recommend it for baking as it smokes in the oven. Foil is also good for wrapping pans if you don’t mind little creases in your bake (e.g. when you’re going to cover it with frosting anyway).
Cooling rack: These wire racks help bakes cool evenly without continuing to bake in a hot pan. Any clean wire rack or surface to allow air underneath the bakes will suffice.
Mixing/prep bowls: A large bowl is always useful so you don’t have to worry about splatter. Most times they sell them in sets so you can choose the perfect size for mixing. Metal mixing bowls are sometimes better if you need to keep ingredients cool/cold because you can stick the bowl in the freezer for a bit before you start. Prep bowls (whether they be actual prep bowls or just other cups/bowls from around the house) help you get sorted before you start so you don’t have to read how much of what you’ll need while baking. I found a bunch of plastic ice cream bowls that I use as prep bowls for smaller ingredients, and I like using melamine/plastic bowls for my bigger prep bowls. Porcelain or metal are also good for making meringues due to the chemistry of plastic retaining fats even after washing.
Mixing utensils: A few things to look for are whisks (to aerate batters), wooden spoons (to mix without having the ingredients react to certain metals and to fold in without knocking out the air you just made with your whisk), and spatulas (to get every bit of batter or softened butter). Sometimes they sell them in sets, so that might be a good place to start! I prefer having smooth spatulas as they’re easier to clean after (versus those with grooves where the handle meets the head).
Thermometer: A digital thermometer is good to have to keep track of the temperature of milk/water (when using yeast) and for seeing whether or not certain bakes are done (e.g. bread).
Ruler: I like the cheap, clear, thin plastic ones because you can use them both to measure and to cut in a straight line. These are helpful if you want to cut squares of your dough for cookies to decorate later.
Pastry brush: I have a silicone one, but natural bristles are good too. You’ll use this to brush egg wash, milk, or other liquids onto your bakes either before or after baking.

The following are nice to have (and sometimes make your job way easier), and may be worth saving up for:

Hand mixer: I like to use a hand mixer for smaller amounts of batters or for frostings/meringue. These are especially helpful in aerating batters, whipped cream, or meringue, cutting your mixing time in almost half.
Stand mixer: Luckily my dad bought a stand mixer, and that thing is a lifesaver! I should be building muscle kneading dough, but instead I just use the stand mixer. It also helps when I want to make large batches of cookies.
Baking weights: Not totally necessary as you can use dry beans or rice as weights. I also see pie chains on sale as well, which would be easier to remove than individual weights. These prevent your crust from rising or puffing up in random places.
Cookie cutters: Most times they’re fun to use, and I like using them to help differentiate flavors when I make multiple batches of the same thing. The best times to look are after holidays!
Pastry bag set: I have a couple (whether it be my own purchase or gifts), but piping frosting is a good skill to learn (which you can practice with your homemade buttercream). Frostings are easier to pipe than whipped cream, and sometimes a shape change in your frosting can turn something plain into something much more showstopper-worthy.
Bread knife: Steak knives are a good substitute for now as they’re serrated as well, but when you make bigger cakes and breads, you may want to think about getting a bread knife as they’re made a bit longer to tackle the bigger bakes.