From what I can gather from your ethos, you seem to care for your fish and are willing to do what's best for him. Nice to see.
That 10 gallon will be a welcome change for your fishey. That 10 should suffice for quite a while, seeing how he's only 2 or 3 inches (? You didn't specify, could you get the exact number, it'd be nice to know). If he gets monstrous, he may need a 15 or 20 gallon as commons can get huge. If that happens and you do need a bigger tank, just think of it this way, you're doing something right.
OK, so getting a bigger 10 gallon tank, check! Getting a filter, check! I'm assuming you're getting a Hang on the Tank power filter variety for the 10 gal. if you're not, that would probably be the best choice. As an avid AquaClear filter user, I vouch for their particular brand wholeheartedly. All you'd need to get is the AquaClear 20 (Mini) model rated for 5-20 gallons. With the waterfall method of reintroducing the filtered water and the surface disturbance it provides, a bubbler will be unnecessary providing you do not add any more fish (yes, do NOT add any more fish, you're stocked to capacity already).
Let me see what else you'd need to know: Ah! Water changes! Water change requirements varies on the circumstances. Say if your water has 500 ppm of nitrate (a MONSTROUS amount of nitrate, normal=40-50 ppm, ideally you would want 0 ppm, but realistically ideal=25 ppm) then I would just change 100% of the water. When you have a cycled tank that's stable in water parameters, then a routine water change regimen would be 25-30% water change every week. If you're starting to cycle, you'd probably want to do smaller, more frequent water changes to help control the ammonia, nitrite levels.
So then this comes to 'how do I change the water'? Just buy a gravel vacuum. For a small 10 gallon, I recommend just buying a self-start mini vacuum. It'll look like a clear, long plastic cylinder. One end will be open, the other end will be sealed with a tube leading into it. All you have to do, is move the tube up and down rapidly until water is forced into the tubing, doing this until a steady stream of water is coming out of the tube. When this happens, just put the open end into the gravel, and shake the gravel (are you getting gravel??) around in the cylinder for a few seconds sucking up all the poop and nasty stuff. If you're not getting gravel (if you want a bare bottom tank) you still want the gravel vacuum. Don't let the name of it make you think you HAVE to have gravel for it to work (that's what I thought
OK, I already told 'ya about the test kits for ammonia, nitrite and nitrate (you ARE an expert on the nitrogen cycle and how it applies to your aquarium now, right??). Now..um, ah, feeding. Be VERY sure not to overfeed. Fish are eternally hungry (I know mine is!). I could feed my little lionhead a 5 pound steak and he'd still be gulping at the surface looking for more food. Feeding once a day should be enough. The feeding should last around 2-3 minutes. They make good flake food for goldfish. Make sure that you buy a right sized container that you'll use all of it up within 2-3 months. By this time, the nutrients in flake food I'm told is notorious for fast losing it's nutrients. So I make sure I use all of it by then. Just write the date you open the container on the bottom. You should also try and vary the diet of the fish for best results. Freeze-dried shrimp, blood worms, frozen fish food, live food, fresh vegetables and fruits (yea, you CAN feed them these, I didn't know this either when I first started out, though mine only seems to like watermelon), etc etc etc.
Another point relating to feeding: Make sure no/minimal food drops to the bottom of the tank. It'll foul up the water and increase the waste level in the tank. A definite no, no.
What else, what else, don't let the temperature of your water get too high. Don't put the tank in full view of a window that lets direct sunlight onto your tank. Your goldfish is a coldwater fish. I believe an optimal temperature for them is 74/75 degrees Farenheight, though I keep mine at 78-79. A noteworthy thing: temperature is directly related to dissolved O2 concentration. Higher temp.=lower 02, Lower temp.=higher 02. I wouldn't worry about that, though, especially if you keep your temperature at the aforementioned.
OK, that SHOULD cover all the bare essentials. I really wish you good luck in the aquarium hobby.