Hi Kelly:)  You've asked a very loaded question, I'm sure you realize!  LOL  I'll give you my opinion, and you can file it with the tons of others that will later be proferred:)  The filter I can recommend right up front - AquaClear.  This is a well made, reliable, and easy to maintain filter; it's a power filter, with its own little motor (please don't think I'm insulting your intelligence, I don't know how much you know:), and it hangs on the back of the tank (or front or side too I guess, if you like:)  I've owned both the smallest and the largest, and really liked them both; my buddy used one of the medium sizes and had good results too.  AquaClears aren't terribly spendy either, which is nice, and all your filter media comes with them - sponge (mechanical filtering), mesh bag of charcoal (chemical filtering), and a mesh bag of biomedia (biological filtering); Foster and Smith has them at a good price.  An air pump I'm afraid I can't help you with, I haven't used one for ages; power filters give a nice little surface agitation on their output, which helps oxygen absorption the same as air bubbles.  Again, I refer to to Foster and Smith (no, I don't work for them:), they have plenty of pumps and can also recommend one for you.  Okay, water conditioning - all I've ever used is a simple dechlorinator/ammonia remover/etc.  This is simply added to your tap water, and neutralizes chlorine, ammonia, heavy metals.  I've used AmQuel brand with great success.  So far as ph, I've never tested, I just add a conditoner to make the water safe from municipal chemicals (chlorine burns a fish's gills, just like our eyes in a pool:); you can get kits from the same place I've been mentioning (I'm quite sure, anyway:)  When it comes to tank size, I wholeheartedly recommend BIG.  Now, I don't mean a giant one that none of us can afford, but the largest you can get a hold of.  I've used smaller tanks, and when your fish are small, as yours are now, it's perfectly fine; but they do grow, and your ryukins could reach 8 inches, 'same as most fantails, orandas, etc. (although some orandas can reach 10 inches); and the general rule is 30 gallons per adult, that is to say, full grown, fish.  Calm down:)  LOL  For your four friends I'd get no less than a 20 gallon, and that magic 30 gallon minimum would be better.  My last two fantails I kept in a 20 gallon tank, and they were happy, but did end up outgrowing it, and luckily I found a good home for them with someone who had a bigger tank.  Another idea, is could could consider what I'm (hopefully) going to be in the process of doing soon - a simple round (maybe rectangular) stock (cattle/horse) tank which is WAY cheaper to buy per gallon of space, and will very easily give 2 (probably just two:) all the room they need to exercise and breathe, and grow to full size.  There's also the half barrel idea, which I've seen successful examples of on YouTube - any non harmful container; and you can use the same filter/air pump you would otherwise (for my stock tank idea, I'm simply going to buy a good water pump, and use that with a length of pvc pipe to make a simple, effective filter).  I should mention too, that goldfish don't need lots and lots of water depth, what they really need is surface area, for best oxygen absorption.  What else...'can't go wrong with TetraFin Goldfish food - it's the standard, and they love it:)  Alright, that's probably too much info, but I do like to help others when I can:)  The catalog I mentioned, is, again, fosterandsmith.com, and they can recommend even more specifically for you, I've always been really happy with their customer service.  My best to your fish:)

If you don't have any live plants then lighting is not required. However if you decide to have live plants in the future, then the plants will need about 10-12 hours per day of lighting for photosynthesis (to make food and grow).
I do recommend you get some sort of aquarium hood with fluorescent lighting since it will make the aquarium look a lot prettier and the fish will look a lot nicer too.  Incandescent lighting is cheaper, but it also consumes more electricity than fluorescent and does not show off the true colors of the fish.

Hi  - I have 4 goldfish in a 40 gallon tank.  I was told not to worry too much about ph, goldfish like the 6.8 and up, which is where my tap water is on the scale.  Buy a ph test from a pet store and test your tap water.  It will most likely be fine.  I read some of the recommendations about filters.  I like the cannister filters more, Eheim.   they do a better job, and the main thing you will have to watch is the ammonia level (get a test for that).  This is created by waste, if you have too many fish in a small tank, that is what you will be dealing with.  The cannister filters do a better job, so I am told.  I have also been told that a larger aquarium is easier to maintain, so staying small may be a little stressful. One thing to keep in mind with filters is you will be replacing carbon, and other pads, this can get expensive every month.  The Eheim filters, only have a pad to change.  It is more expensive, but in the long run you will save on supplies.  Spend the $$ up front or you will be spending it on supplies.

Kelly,
You could always get an aquarium kit which has everything included.  A deluxe kit is usually a bit more but includes the floresent lighting rather than the incadesent.  I prefer the floresent because it doesn't heat up the aquarium and it cost much less to use, liuke someone already mentioned.
I just bought an aquarium deluxe kit by Aqueon.   It comes with a power filter, hooded light, florescent, and condtioner etc.  I added a bubble wand and air line and pump and was good to go!
There are many aquarium kits out there by other companies too.