Hello there! I recently got a goldfish and he (not sure about the gender) was named Yoshi by a friend of mine. I have never had a fish as a pet before, or any other animal, so I would appreciate any advice for caring for little Yoshi.

He's such a tiny little guy, I have lots of fun just watching him swim around in his bowl

I've been reading up on how to care for goldfish online these past couple of days and have read many articles that are leaving me with more questions than answers. Like how it's probably not a good idea to raise fish in a bowl or about what to feed them/how many times to feed them.

He looks like he's breathing a bit hard--I cleaned out his bowl this morning and I think an aerator is what I should buy next?

Looking forward to getting to know everyone and their cute fishies 

Pretty much what Hanna said Once you answer those questions we can help you alot more and give your fish the best and longest life possible.
Start researching on the tank cycle as it is one of the most important things you should know. There is plenty of information on it on the internet and on this site, so have a look.
It doesn't sound very good so far, it sounds like he is living in a 'normal' goldfish bowl which is actually not good at all and by the way you said it also sounds like you cleaned out the entire tank which you should also never do (all to do with the cycle).

I would absolutely encourage you to buy a 20 gallon tank, since this fish is a streamline shaped variety, as they are faster and need more space to swim than a fantail or other egg shaped kinds.

You also need to make sure that you've got sufficient filtration, meaning 5 times the tank volume at a minimum (The more the better, always when it comes to filtration). So you need a filter that turns at least 100 gallons per hour, it's the same with filters as it is with tanks, try buying something big enough from the start so you won't have problems later. An external filter is ideal when keeping goldfish, since they are very strong, and have lots of space for filter media

Other than that, get yourself three different kinds of high quality staple pellets, some floating and some sinking, just for a varied diet. This needs to be supplemented with live foods and fresh vegetables. Frozen or freeze-dried blood worms or tubifex worms are among the most common ones, but there are also brine shrimp, krill etc available.
Fresh vegetables include peas (popped out of their shells), cucumber, lettuce, boiled carrot, zucchini... etc, just be sure that they're free of salt!

Now to answer some questions:

Let the new tank stand at least for 24 hours for the temperature to stabilize. And add some beneficial bacteria. For my tank, I used "Fluid Filter Medium" by Easy Life, it's a very good thing, as it prevents diseases and strengthens the immune system during this sensitive time

When you change the water, change no less than half, 75% a week is even better And buy a special siphon, so that you can clean the gravel at the same time, siphoning out any waste that has accumulated between the rocks. You can also wipe away algae and dirt on the tank walls, but leave some growing on the back, the goldfish love eating some algae now and then

WATER CONDITIONER IS CRUCIAL. Go buy it right now!!! The chlorine in your tank water is hurting your fish terribly! That's just what it's for, to neutralize the chlorine and make the water "fish-friendly". Without it, your fish is suffering, being burned by the chemicals :/

Just to clear one thing up Nossie it is just a 75% water change until the tank has cycled. Then you can go down to 10-25% depending on your water quality as of course it changes
Try to get the biggest tank you can afford even if it means borrowing some money or going without somethings for awhile as a bigger tank is vital for this fish's survival, it can not live in this bowl. Though you should be aiming for 20 gallons, I think you should also be fine with a 10 gallon one for one goldfish! I think it would be best to put your goldfish in the tank as soon as possible well after 24 hours like Nossie said. Put the water from the bowl in the tank and buy beneficial bacteria to speed up the cycle.
It is just vital for you now to get this fish out of the bowl but once it is in the tank to keep a close eye on your water. Do the water changes every second day for now and get your water tested so you know where to go each day.
Once that is all sorted, you can do a small water change (10-25%) per week or fortnight. Also wash half of the filter floss/wool in dirty tank water according to how dirty it gets. Only wash half at one time and change only half when needed.
Feel free to ask more questions. Just get the important things sorted for now I would say.

Following on from Nossie's last point, invest in a water pump so you aren't stuck lugging buckets around. Just make sure you can reach a drain and a faucet.

Ohh I see  But in such a big tank how do you manage to do 75% water changes? Mine is only 20 gallons and I couldn't imagine doing that much. Unless you do what Jujube said?

I can't believe I said gallons, oh no I'm turning into am American

But yeah I meant your tank Nossie. My gosh that must take such a long time, it was hard enough filling my tank up with buckets, my whole body was shaking by the end. Your such a good fishie parent

Yep I have a 10 litre bucket too. Another problem is I fill the bucket up in the laundry which is down the other side of the house because the bucket gets suck in any of the other sinks in the house! Learnt that when I was filling the tank up and water went everywhere!!

Do you speak Swedish? I always thought you were from Finland 

Andrea- just had to give you karma for saying you're turning into an American.  Not a thing wrong with that.
Oh, one other good thing to get is a little net for scooping the tank.  It's good for cleaning out floating stuff like extra food scraps or poo.  And also make sure your fishies have a decoration or two to hide in.  You may want to see your fishies, but they like their privacy sometimes and need a place to chill by themselves.  And I think it goes without saying that you do NOT tap on the tank or the bowl.  The noise scares them.  Follow the suggestions on this site, and you will have happy, healthy fishies in no time!

Ok, I've kind of transferred ownership of Yoshi to my brother and he's renamed him "Mario." Haha But I got another goldfish--it's white with a few orange spots...kind of like a koi? I haven't thought up a name for him (genders!) yet.

Oh, and I've gotten them a tank and this time I had the water conditioner to clean it with. I put in the drops before I put them in the tank, while I was waiting for the water to cool down to room temp and then put the fish in. But for future reference, I wonder, do I have to take the fish out before I put in more water conditioner?

My camera's not taking very good pictures because it's dark outside right now, but I'll get pictures up soon.

Does your brother have a proper tank for Yoshi?

He's probably a comet.  Our Izzy is colored like that.
Andrea- never been to Finland but I was an exchange student in Oz back in '96.  We were mostly on the east coast, Sydney up to Brisbane, and then to NZ.  Australia is lovely.  We went during June and July, which is our summer but your winter, but my mom didn't get that and insisted I pack shorts and light shirts and leave my coat.  As a result I remember being quite cold while I was there.