Here is what you will need before you get fish. I try to make this as clear cut as possible but of course, your situation will probably vary at least slightly. This will be for a cookie cutter 20 gallon. Note, some of these components are available in a package available at walmart, petsmart or petco ect so go with that if it includes most of these at a reduced price. Here we go.
1. Aquarium- You can save good money here by going with a package. Go as big as you can but most beginners go for 10-29 gallons. As you'll see, it gets less expensive per gallon as you go up so that's one incentive to go big. Larger tanks are in some ways easier to maintain because a larger volume of water fluctuates slower than a smaller one; more room for error. More water means more fish or it means cleaner water. Of course, changing water and scraping algae will take longer on a larger tank but it's not bad considering you have better options fish wise. But I do understand if you're not totally into keeping a 4 or 6 foot long tank if you unsure about the whole hobby so you may want to start smaller. Petco also has dollar a gallon sale on tanks 10-55 gallons once or twice a year. You can also find good deals on craigslist mostly for larger tanks but expect to spend time cleaning it up.
2. Filter - You will probably end up going with a standard hang on back power filter. Go for one rated higher than your tank. Example, you have a 20g, get the one rated for 20-40gs rather than the one for 10-20g. If you're going to splurge on one thing, make it the aquarium or the filter. Often times, a good filter can increase potential fish stock or decrease maintenance. Popular brands are aquaclear (A+) , marineland (B+) topfin/penguin (B). Packages usually come with penguin/topfins but they are usually good enough for a moderate stock as long as you maintain it. To start, get 1+ set of replacement filter cartridges, they will last months. You can add a filter later on also to supplement. Canister filters are expensive but are better in many cases than power filters. I won't get into sumps here. Sponge filters are better in some cases as well but still go for the standard hang on back power filters. Power filters can range from 15$ (10g) -80$ (55).
3. Heater- There is a 95% chance you guys will start with tropical fish and will thus need a heater to get the water to around 78 degrees F. Just get one rated for your tank. Getting two less powerful ones will provide better insurance in case one breaks but usually this isn't necessary unless you're dealing with harder fish that can't survive a day in cooler water. I would say avoid topfin heaters if you can, but they usually come with the package so you may just need to make it work. Read reviews for the heater you're going to go with because I have to admit, I don't have great luck with them and I don't really use them as my fishroom is heated itself.
4. Lid/lighting- Fish are fine without lighting but it makes the tank look so much better so I wouldn't consider an aesthetic tank complete without one. Most fish are good without lids as well. Again, the packages usually come with a decent hood with a fluorescent strip or even compact fluorescents. Hoods can be expensive so usually here the package will save you cash you can spend on other stuff.
5. Substrate- The beginner will probably go with the standard gravel. I'm pretty sure the bag will say go with one or two pounds per gallon but that isn't too accurate because of height differences. It doesn't have to be too deep, a layer barely cover the bottom can make it easier to clean too. Many of my tanks are bare bottom but otherwise, I usually use washed home depot sand. I'll probably do an article on that later but for now, go with the regular gravel at 1lb per gallon. Expect to pay less than a dollar per pound.
6. Conditioner- Seachem Prime is the conditioner to use to rid your tap water of chlorine and chloramine although others work too. Bottles start at 3$. It can be dosed in emergencies to get down ammonia as well. This should be one of the ONLY chemicals in your arsenal. Don't listen to anyone trying to sell you snake oil that will make your tank perfect with no maintenance. The only other chemicals I would consider are some meds.
7. Siphon- A vital tool that too many aquarists don't have. You use this to siphon debris from the gravel. This gets rid of uneaten food as well as removes water into a nearby bucket. A medium one costs maybe 8$. I would definitely recommend purchasing a python though for maybe 30$-40$. You hook one end to your sink and the other into the tank. This way the water gets drained right into your sink and you refill the tank right from your sink. Best 30$ I ever spent as a beginner. No more carting 5g buckets.
8. Water- It hurts my eyes when I see people buying betta water or store prepared water in general for their basic setup. Conditioned tap water will work in almost all cases.
9. Food- You will need to know what type of fish you are getting before you buy food of course. Basic flake food is good for most community fish. Sinking pellets are good for catfish, loaches and bottom dwellers. I'll get into pleco diet later. Variety is important so be sure to mix in random foods like frozen/freeze dried bloodworms, mysis shrimp, brine shrimp, tubifex worms ect.. It's good to start out with a nice assortment and using them a couple times a week will make them last forever.
10. Ornaments- Not really essential but the tank would look odd in the living room without anything. It does depend on the fish, but I do like to keep things as minimalistic as possible. Go for some fake plants and maybe a couple smallish structures. Also, don't get live plants just yet.
11. Cycling materials- *Ding ding* here's cycling again, very important. You will need to choose a method to cycle with and get the stuff for that. Basically check out my cycling article on how to do that, will write soon.
So there you have it. Remember don't waste your money on chemicals other than the ones specified. You'd MUCH rather spend money on the best filter/heater as possible than ornaments. More updates soon.