Hi Fuwball and welcome to the forum,
It's terrific that you've gotten a bigger tank, 10 gallons is faaar better than a 1 gallon tanks but please keep in mind, you will more than likely have to upgrade once the fish has grown
Common goldfish can and will grow up to 1 foot (sometimes more) when given proper care, usually they are kept in ponds, so when a common goldfish reaches around 5 or 6 inches, that's *from what I've read* the best size to move them at. Or alternatively you can keep them in really large tanks of 6 foot in length (or more). They need lots of space purely for growth, exercise and good health
I completely understand your situation though, as I'm in a similar predicament. I previously had my goldfish Lucky in a 5 gallon tank and when I learnt of my mistake upgraded her to a 23 gallon tank/3 foot in length. I will be upgrading her to a pond much later as she's a bit of a runt and is slightly stunted from being kept in her small 5 gallon tank for so long, so she grows real slow, it's taken her most of this year to grow half an inch
Just wanted to share that so you don't feel alone we all make these mistakes
Okay now as for your questions;
Air stones; Theses are mainly used for circulation in a tank, now normally if your filter is good enough it will provide enough circulation for you and you won't need it, however in some cases it might not and these guys become useful in those situations. For example my filters output is rather strong, so my spray bar faces the back of the tank to create less current, but at the same time this doesn't do much for the circulation, so I have air stones in my tank. They are also used for decoration too. But they are not necessary pieces of equipment!
Filters; For goldfish, I would suggest an external filter, they are much bigger than any internal filter and as a result it means more filter media = more bacteria can grow (bacteria in the filter will deal with the ammonia/nitrite in the water).
Depending on where in the world you are... I've been using a filter made by a company called All Pond Solutions, but I believe in America it is called Aquatop (basically one company somewhere makes them, various companies sell them).
They look like this;
They are the cheapest brand of external filter I've seen in the UK and they can come with a built in UV bulb.
I made a review of mine, so be sure to take a look - (Link is not visible to guests. Please register to view.
and of course do your own research as to what your budget can manage and what filter you'd prefer
There are a lot of options out there.
With regards to setting up the tank, I would suggest you grab a liquid test kit (not the strip tests, liquid will last much longer and is much more economical
I've had my liquid test for over a year now). This will help you know when the tank is cycled. Now I would go into more detail but I have some training to be at, so when I get back I shall explain more about fishless cycling, unless one of the others beats me to it
Fish-in cycle could be a possibility here, given the circumstances, but I'd like to see what the others think of that idea before suggesting it!
Oh and also, welcome to the world of fishkeeping