Thanks for the advice.Just to let you know that I inherited the fish tank from a housemate that left it behind. I am a newbie and could not just get rid of the fish.
Well then bless your heart for not flushing them!! I know people in your position who did just that. Good for you.
Now as far as the deaths, they are unfortunate considering that the person who had the tank before you obviously didn't know beans about what he was doing. That is not your fault. I'd like to know what your expectations are for continuing to keep the fish. Because this has been thrust on your hands, it's not like you were prepared to have to invest so much into caring for a couple of goldfish, right?
If you do decide to keep goldfish for that tank, yes... you'll have to invest in a larger aquarium. Goldfish are very messy and need lots of filtration and aeration to live happily. Hence the whole "10-gallons per fish" rule. That ensures that the goldfish has enough growing room, enough space to produce and meet his oxygen needs, and also enough water to handle the immense amount of poop goldfish tend to make. Because who ever likes to swim in waste-water, right? ^__^;;
Some items you might want to invest in if you decide you want to keep the surviving fish:
Liquid Freshwater Master Kit.This handy piece of equipment will help you monitor the pH, ammonia, nitrAte and nitrIte in your aquarium. All of these must be at a certain balance that is necessary to reduce the chances for illness, distress, and death in your fish. A lot of problems can stem just from one or all of these being out of whack. (as you'll learn through reading around this forum)You'll also want to buy a few buckets specifically for cleaning and maintaining your aquarium, a gravel vacuum if you don't have one, and also... eventually a 30-gallon setup or larger to house your goldfish in.
If this seems like too much hassle, that's understandable. If you want to just keep the 16 gallon tank, perhaps it'd be best to give the goldfish to your local fish store (LFS as we call it here), scrub down the tank. Replace the filter cartridge and invest in a heater. There are tons of tropicals that can happily exist in only 15 gallons.
However, if you want to stick to the task of keeping your fan-tails, save up for that 30-gallon and feel free to ask questions of us!! We'll help you however we can.
Oh, and welcome to the wonderful (if sometimes demanding) hobby of fish-keeping!! ^___^