They are very small goldfish. I see goldfish kept in a bowl and our tank is much much bigger than that.
Fair enough, but where any advice stems from here on the forum is the fact that 'small goldfish' become large goldfish. When considering stocking levels we always take into account the size of a tank and match that to the maximum average size of a fish that it will grow to. You've got to think of the future and is a common mistake not taking this into account when first getting into fishkeeping.
To this end I will also say: get rid of the Goldfish! True, they need a lot of room, are cold water and are the dirtiest of fish!
With those gone, I'd say that for your 10G tank that the remaining stock you will be left with is all you'd want to put in it & no more.
12 neons would look great! BTW, neons are fine with a Betta....
10.5 gallons and there is a filter pump and separate pump for oxygen bubbles. Not sure if this is cycling.
Oxygen bubbles are there purely for decorative purposes. Consider this as another thing your friendly LFS has sold you that you do not need. But they do look nice so whatever.
& NO, this is NOT cycling (I can see how you think this though)! OK, where to start..... in a nutshell then:
Fish pee. They pee Ammonia. This is poisonus to fish and needs to be removed from the water using a filter.
OK so far...
The (blue) filters hold bacteria that chew up the ammonia and convert it to a less poisonus chemical (Nitrate). The green filter then converts Nitrate to the even less poisonous chemical Nitrite.
(Nitrite levels & indeed all chemical levels are then further reduced via
weekly water changes)
Good o!
BUT - your blue filter does NOT come with any bacteria in it. You have to cultivate a level of bacteria BEFORE adding any fish. This is called cycling and is done by adding small amounts of pure ammonia to a tank
without any fishin it. This process usually takes about 6 weeks.
IF you have a good LFS you can 'seed' your filters to speed up the process in which case cycling can take only 2 weeks.
There are pinned topics on this very important subject....
Andy
PS
I'd start looking for a better LFS whose staff are better educated (or least not simply trying to flog you unsuitable fish!)