I keep fish and we do some aquarium maintainance for a few local people who can't do it themselves...
Cold water fish like goldfish need VERY large tanks and heavy filtration - the common goldfish has an adult size of 6 to 12 inches (in reality the adult size is 8 to 12 inches but the average is dropped down by the number of significantly stunted fish due to poor housing).
There is an oft repeated myth that fish will limit growth to the size of the tank - I mean this is sort of true... in the same way that if you kept a small child in a box full of their own piss and shit they would also be pretty stunted... it's not a good thing! Fish will absolutely grow so big they can't comfortably turn around in a small tank, but most will die before that happens... painfully, (because yes they do feel pain!) probably from ammonia burns to their gills.
The natural lifespan of a common goldfish is around 30 ish years... probably longer, but again the number of goldfish fry and juveniles dying at a year or two old brings it down.
Go as big as you can possibly accomodate - focus on keeping water quality perfect. The bigger the volume, the easier that is to do.
Look at what sort of water you have - fish are either evolved to live in hard water or soft - a soft water fish cannot survive to its full potential in hard water, and vice versa (and trying to make them do so is to either have their bodies bloated with water they can't shift, overloading kidneys, or the minerals stripped from their bodies that they can't replace, as the wrong water messes with their ability to osmoregulate).
YOu can alter the water if you want - I live in a hard water area and keep soft water fish. I have a Reverse Osmosis unit and have to adjust the chemistry of every bucket that goes into the tanks - you probably do not want!
Some small fish need a lot more room than you realise - for example, neon tetras need to live in big schools and have a big tank, they swim a lot!
A siamese fighting fish or betta however, bobs around, likes to explore but would be more than happy with a 10gallon tank with some nice broad leaved plants to sit on.
Tanks (whether cold or tropical, hard or soft water) need to be set up and run so that the filter 'cycles' this means it builds up a colony of denitrifying bacteria.
That process is started by you adding ammonia, and monitoring it, and it takes around 6 weeks, which is very boring for a kiddo.
Couple of alternatives (but not the 'instant bacteria' products people try to sell. They don't work, wrong bacteria).
Set up a filter running in a big bucket somewhere out of the way. Run the cycle just as you would for the actual tank, it is the filter you're cycling not the tank/water anyway - theres tons online telling you how to cycle a filter, look for 'fishless cycle'.
When you get the tank set up, planted up, filled with dechlorinated water etc - and water up to temp, pop the now cycled filter in there and get that running. You need to maintain the cycle by feeding the bacteria, so you either continue to dose with a bit of ammonia until you can put in some fish, or just put in fish. Do not overload it by fully stocking the tank straight away though!!
This does still take around six weeks, however you can often set this up somewhere without a kiddo seeing (if its for a tropical tank you will need a heater in the bucket too!) - so you haven't got six weeks of 'im bored can we get a fish today?'.
The other option requires you to know someone with a nice big healthy tank, and them being willing to let you s hove your filter on their tank for a few weeks, pop a few of their rinsed but still bacteria laden sponges from their filter... and then transfer to your tank (or just nick some of their filter media) - it need stheir tank to be VERY healthy, no parasites, no disease...
You also need to do the transfer pretty fast as the bacteria die off if starved of oxygen/temps drop too far.
Finally - most fish shops will tell you a load of old rubbish, particularly Pets At Home who often don't even know what species they're selling and will tell you to just let the filter run the water round the tank a few days then stock fully, or will ask to test a sample of water ... Then they test for ammonia which any tank full of clean tap water will not have, declare it safe and sell you fish (which then poo in the water, ammonia builds up and within a week your fish are dying of ammonia burns).
For a kid into fish..
Small space - 35 to 45 litre tank would suit a handful of shrimp and some white cloud mountain minnows - they both want medium to harder water, and a heater keeping things at around 20c will suit both too.
Alternative - same tank, single male betta/siamese fighter.
Alternative - soft water - loads of chilli rasboras
Alternative - hard water - endlers guppies (but get just males or you'll have MILLIONS of babies)
If you can go up to 250l, you could keep some of the more common tetras (soft water)..
Asides from endlers guppies and fancy guppies, theres not many small fish for hard water tanks - mollies and platys actually get fairly big and will want a 200+ l tank to have more than a pair of either.
If you can go REALLY big 400l+ then the hard water world opens up with all the african lake cichlids which can be very bright (but a lot of these are very aggressive).
Keep in mind that a large tank, whilst taking a little longer to do a water change (around 40% weekly) will be much more forgiving of you being a day or two late on that change than a small tank! Also much easier to leave for a week or two whilst you go on holiday without needing to have someone come fiddle with it!