Basic Freshwater Aquarium Principles; Filters, Heaters, Bio-cycling, Live Plants, Feeding, Cleaning and Chemistry

AQUARIUM:
Start with as large an aquarium as you can afford. There is a very BASIC priciple (this only applies to beginners), that is to have 1-2 inches of NARROW bodied fish per FILTERED aquarium gallon. Goldfish are dirty and fatter, so I would triple this with them (3"). This also only applies to a standard rectangular aquarium.

Obviously longer fish need more tank width and length. I would decrease the amount of fish proportional to the gallons in a tall aquarium or hexagon aquarium.
Remember, many fish purchased can grow much larger than your original purchase size (ex: goldfish), so keep this in mind too.

FILTERS:
I always recommend two filters minimum per aquarium for redundancy and for improved biological (denitrifying) filtration. For a small aquarium, a combination of a hang on the back and a sponge filter. Or a sponge filter and an internal power filter. You want to make sure and rinse your sponge or cartridge out in used aquarium water to maintain your beneficial bacteria for bio filtration.
Other filters of note include canister, wet/dry, under gravel, and fluidized bed.

There are four types of filtration:
Biological; the removal of nitrogenous waste (ammonia, ECT), which is the most important type.
Mechanical; the removal of larger debris (organic and inorganic) before it can go through the nitrogen cycle (organic)
Chemical; The removal of chemical contamination via carbon, zeolite or many other products. This becomes less important in a healthy, established aquarium.
Germacidal; The use of UVC or ozone to kill disease pathogens and control the Redox potential.

HEATER:
Most tropical fish do well at a temperature between 76 and 80 degrees Fahrenheit. (Discus prefer warmer).
Goldfish do not need a heater.
I recommend 25 watts for every 10 degrees of ambient temperature you need to raise your aquarium temperature. EX: If your home is 68 degrees and you have a 40 gallon aquarium, to reach a temperature of 78 degrees you would need a 100 watt heater.

BIO-CYCLING:
Your aquarium will not be at peak biological filtration for 6 weeks (or more). To start your biological filtration, there are many cycling products available, such "Cycle" by Hagen. My success with these products is mixed at best, it is very difficult for the aerobic bacteria that are needed for cycling your aquarium to live in a sealed container kept at room temperature, as they die very quickly without oxygen.
I prefer to add gravel and/or used filter sponge or cartridge from another aquarium.
This method of adding media is much faster (you still have to take it slow), and provides all the necessary bacteria, the only negative is adding disease pathogens to your aquarium, but I have rarely encountered this problem.
We used this method for our Aquarium Maintenance route for years and never lost a fish to Ammonia or nitrite poisoning.

Another method is fishless cycling where un-scented ammonia is poured into the aquarium (3-5 drops per gallon pure ammonia) so as to bring your ammonia level to 4-5 ppm. Then it takes about 3-8 weeks for the aquarium to cycle. Although this method is growing in popularity, I do not recommend it, not because it does not work (it does), but because human nature is to want to add fish sooner than the 3-8 weeks it takes for this method.
The method of adding media is much faster (you still have to take it slow), and provides all the necessary bacteria, the only negative is adding disease pathogens to your aquarium, but I have rarely encountered this problem.

Cycling is what is referred to as the Nitrogen cycle. Waste (nitrogenous) from the fish is broken down first from ammonia (NH3, the most toxic) to nitrites (NO2, less toxic) to nitrates (NO3, least toxic- but high amounts can stunt fish growth and lower disease resistance).
At a pH of 6.5, NH3 (ammonia) converts to NH4 (ammonia) which is basically non-toxic to most fish (many ammonia removing chemicals to a similar ion change, as they do NOT actually remove ammonia). If you have plants in your aquarium they will directly consume the ammonia (especially hornwort), thus rendering the NO2 (nitrite) part of the nitrogen cycle null. The danger here is if your pH climbs above 6.5 the ammonia can change to much more toxic NH3 and the aerobic bacteria needed for nitrite consumption will be sparse.
For more about th nitrogen cycling, see this article: AQUARIUM NITROGEN CYCLE

LIVE PLANTS:
Live plants are desirable in my opinion, but many artificial plants can look quite realistic when properly arranged or used in conjunction with live plants. For a beginner live plants are more difficult, but not a lot.
The benefits of live plants are they are great at nitrate removal and keep a natural balance to the aquarium, removing CO2 and adding oxygen (only during daylight). Hornwort is an excellent plant for nitrate removal (even ammonia removal), and is relatively easy to grow. Banana plants (when available) are also a very easy plant.
Here is a list of "easy" beginner plants:
Compacta swords, hornwort, Red wendtii, spiral valis, dwarf subulata, Hygro, Java fern regular or lace.

Be careful with many fish that will "mow down" your plants such as: Silver Dollars, most African cichlids, and even goldfish.
For healthy plants I suggest a substrate of #00 sand mixed with laterite about 3-5 cm deep with a layer of #3 gravel on top about 2 cm deep. This combination works well for plant roots, ease of vacuuming the top layer ONLY (where plant roots are), and for better bio filtration. You can substitute laterite with a sandy top soil (although usually not as good a source of iron), by preparing the soil thus; Gather sandy top soil, add water with a 10/1 bleach solution, mix for a couple of minutes, then rinse (with a de-chlorinator for first rinse) until the water runs relatively clear. The sand that is left is what you mix with your plant roots.

FEEDING:
I recommend feeding high quality fish and plant based foods. Quality ingredients include: spirulina, fish meal, FD Brine Shrimp, shrimp meal, Vitamin C & E, lobster shell.
Fish cannot digest proteins from beef well, and fish get most their energy requirements from fats. Some quality foods include: Omega, Spirulina 20, Ocean Nutrition, Hikari, Sanyu.
Feed you fish two to three times per day what they will consume in three minutes.
Feeding foods high in poor quality proteins can increase your nitrate levels, as an essential ingredient in protein is nitrogen, and if unusable by the fish, it is excreted, entering into the nitrogen cycle.

CLEANING:
You should try and have a schedule of changing 20% (or more) of your water every week. I recommend using a gravel vacuum, you need not remove the fish while using a gravel vacuum. Make sure the water you add back in is the same temperature and ph, and has no chlorine or chloramines.

CHEMISTRY:
Keep your ammonia level at 0, your nitrite at 0, your nitrates below 20-30, and your KH above 80 ppm. Ph depends very much on the fish you are keeping. Discus prefer under a ph below 7.0, while Mbuna African cichlids prefer above 8.0
A very general ph of 7.2 -7.5 works for many community fish. Crushed Coral and/or Wonder Shells can help maintain a high pH when you desire an aquarium with a higher pH, KH, & GH, especially where tap or well water is very acidic (Wonder Shells are much faster at dissolving to the desired KH and add electrolytes crushed coral does not!).
For a lower pH in aquariums where the tap water used is very high (usually 7.8 or above), I have used blends of RO (Reverse Osmosis) water and tap water. The ratio varies with the tap water pH, KH, & GH and the water conditions I want to achieve. With Discus it can be as high as 75% RO. Then to maintain these conditions I use peat in my filters. Note that GH does not affect pH, and magnesium (a major ingredient of GH) is important to fish metabolism. Also note that calcium which is the main factor in KH (which does affect pH!) is also important for fish metabolism and fish health and healing. With the above method of using RO (or DI) water in a blend with tap water and peat, I have still been able to maintain a KH above 80 ppm (for proper calcium absorption), sometimes with additives such Wonder Shells or Calcium Polygluconate.

CONCLUSION:
This is only very general information, there is much more in depth articles about each of these subjects and more available. But these are sound principles to follow, based on 27+ years of aquarium maintenance experience. As new and better methods become available, I regularly update my information to reflect this.
For my more in depth full article:
Aquarium Information