Hi and welcome!
Pixieil, do you know what the pH, kH and gH of your tap water is? What is the problem with your water that makes it "just awful"? I'm not sure how much you know so I'll start at the very beginning.
Here is a quick rundown of the equipment that you will need:
-Test kits for ammonia, nitrite, nitrate, pH, gH and kH. The first three are most important along with the pH.
-Water conditioner that neutralizes chlorine, chloramines and heavy metals. I think that the stress coat that you have does this.
-Net
-Thermometer.
-Heater. Do you know what kind of goldfish it is? Common and comet goldfish are coolwater fish and a temp in the 60's would be fine. If the fish is a "fancy" gold then a temp in the low to mid-70s is fine.
-Filter
-Gravel
-Plants (fake are fine-goldfish will eat live plants)
-Tank and light
I have bought a 10 gallon tank, but I dont know how to set it up
Was the tank a kit? Did it come with a heater, thermometer and filter? If not, you will need a 100w submersible heater and a filter that has the capacity to turn over at least 10 times the tank volume. Thermometers are easy enough, they have a type that suction cups to the inside of the tank or the types that stick on the outside of the tank on the glass.
To set a tank up you need to wipe the inside of the tank with a wet paper towel and then fill it up and let it set to make sure that there are no leaks. This normally isn't a problem with new tanks but it's always better to be safe than sorry.
After you ensure that the tank isn't leaking it's time to set it up. The first thing that I do when doing this is to attach a background the back of the tank. I trim up black plastic garbage bags for this and attach them using clear tap. Pet store sell backgrounds for tanks but these will cost a few dollars. You could also paint the outside of the "back" of the tank with a blue or black paint.
Once the background is attached I turn the tank right side up and add a couple of inches of aquarium gravel after it has been rinsed off. Avoid the larger gravel and stick with the smaller 2-3mm gravel. This smaller gravel gives the bacteria more surface to colonize but we'll get to that later.
Next, fill the tank up and add the correct dosage of water conditioner. Don't add the algae killer. The only chemical that should be added is the stress coat. After the the tank is filled you can place the heater into the water but don't plug it in for at least 15-20 minutes otherwise there is a risk of it shattering. You can also setup the filter and turn it on. Let the tank settle and run for 24 hours or so to make sure that everything is working properly before adding the fish.
Cycling
Cycling is the process of developing nitrifying bacteria in the filter and on every surface of the tank. This bacteria is what is responsible for the biofiltration of a tank. Once established, this bacteria will convert ammonia to nitrite. Another bacteria will convert nitrite to nitrate. Ammonia is highly toxic to fish. Nitrite is less toxic than ammonia while nitrate is the least toxic of the three compounds. Normally, with a regular maintenance schedule, nitrates can be kept around 10ppm without much difficulty.
Here is what you need to know about cycling:
The cycle is started when fish are added to a tank. Fish give off ammonia from their gills through respiration. Fish also pass a dilute ammonia in their urine. Leftover food, decaying plant matter and other waste products will also give off ammonia. All ammonia is toxic at .25ppm or above. The amount of ammonia that it takes before a fish succumbs depends on the species. In a cycled tank there will be 0 ammonia.
It can take up to two weeks before enough nitrifying bacteria is established to bring the ammonia down to 0. Once this biofilter is established, it will remain in the tank and grow and decrease as you add and lose fish.
The bacteria responsible for the biofiltration is aerobic. This means that it loves and needs oxygen to survive. The majority of biofiltration in a tank takes place in the filter. This is why it is important to have a good quality filter with lots of room for sponges or filter floss. This will give the bacteria a larger surface area to colonize. The bacteria will also live on every srface inside the tank as well. They can't be uprooted by gravel vacuuming. The following is what can kill the bacteria:
1.Chlorine- never rinse your filter media in tap water. This will kill off the bacteria colonies. Rinse the filter cartridge, sponge, floss or whatever you are using in a bucket of used tank water and reuse it.
2. pH swings
3. poor filtration
4. temperature changes
5. daylight/light- this why filters are dark
6. medications such as antibiotics (regardless of what the packaging says)
After the initial two week period you will see a climb in nitrite. While nitrite isn't as toxic as ammonia it is still toxic-usually at 2ppm or above. To counter nitrite toxicity, add 1 teaspoon of aquarium salt per 20 gallons. Table salt is fine to use but it must be non-iodized. Water changes and gravel vacs must also be done to remove high nitrite levels.
Nitrites will remain very high for about six weeks. The bacteria responsible for converting nitrite is a painfully slow growing one. This bacteria are also sensitive to the things listed above.
One day you will find accumulating nitrate with falling nitrite. When the ammonia and nitrites both have a reading of 0 and you have a noticeable buildup of nitrate the cycle is complete.
Once complete, a regular maintenance schedule of weekly 20-30% water changes should be done along with thorough gravel vacs to remove fish poo, leftover food and other waste. This should be enough to keep your nitrates between 10-15ppm. If your nitrates are higher then you need to cut back on the amount of food that is fed daily or increase the frequency of water changes. The lower the nitrates the more your fish will thrive.
HTH some, don't hesitate to ask when you have more questions.
Bryan