Small goldfish tank is an oxymoron. With fancies you are looking at a minimum
of 20g per fish or 55g for three. An with shubunkins and comets you can pretty much triple that. Quadruple it even. Goldfish are dirty, dirty, dirty fish. They produce a lot of waste from those convoluted stomachs!
The goldfish you see in stores are actually only fry. The smallest goldfish variety will reach an adult size of 8 inches, with most fancy types reaching 12 inches. Double that for comets. Furthermore, they need heavy filtration and tons of oxygen. Given proper care and space, however, a goldfish can easily live for 20 years.
Most pet stores would also tell you you could do white clouds but to be fair they need a group of 6 to be happy and are so active that they are better suited for a 20g minimum.
3g, however, would be an awesome size for a betta, provided you get a heater. They actually prefer warmer water than most tropical fish, but small heaters aren't that expensive. Plus, bettas are so nice and colorful, full of personality (you laugh now...) and
are solitary so you have a good excuse not to take on any more fishies! When properly taken care of (ie NOT starving in a dirty peace-lily vase) your betta could be with the family for up to 7 years!
Here is a really good care sheet to check out: http://www.tropicalfishkeeping.com/b...are-faq-20058/
And a little mythbuster on why bettas need at least 2g to live: http://www.tropicalfishkeeping.com/b...ive-tiny-8923/
By the way... welcome to the fish forum!