well, it depends on your specific situation what will work or wont work.
Lets first start with your case. Let's assume you're like most people and keep your tower at your desk on floor or on a lower shelf. If you want a quiet computer, it start with your case.
Your Antec 300 case is nice, but its not known for being a quiet case. You have several fan slots, none of which seem to be covered by default. This allows a lot of sound leakage. (I've been there, had the classic Antec 900). So not only will you hear case fans, you'll hear almost every other component in your system as well (GPU fans, Harddrives, and anything else that hums/moves).
So Lets say you're dead set on your case (not my recommendation... usually cheaper/less hassle to replace a case than to mod one, usually have sound dampening and have indirect sound pathways), You first need to determine your air flow and number of fans you want to use. My personal recommendation is no more than 4 case fans (At most) and 1 cpu/heatsink fan. In your case, I would use 1 rear exhaust fan, 1 side panel intake fan, 1 front intake fan (upper), and 1 HSF.
As I haven't a clue which Heatsink you're rocking or the amount of internal space you have (due to components), I'll assume you have lil to no room for fan shrouds. So not only will the fan choice matter, but other things as well. Now, if money is no option, I'd recommend 3 of these (Noiseblocker NB-Multiframe M12-PS 120mmx25mm Super Ultra Silent PWM Fan - 600-1500 RPM) - FrozenCPU and other places carry them. They do cost about $30 a pop, so its not cheap, however their mounting design really helps with vibration. Corsair stole their design and incorporated it into their own fans. (which can be found for a bit cheaper). You want fans that decouple the motor from the mounting... while rubber silencer CAN help reduce vibration... you typically have a screw that connects the fan to the case... which still transmits vibration. I still have to replace one of my case fans with noiseblockers/corsair fans. They are worth the investment.
For your HSF, if you want to change that out (trust me, you do if you're asking about quiet fans). I'd recommend a Gentle Typhoon (GT-14) if you want a continuous speed fan that's quiet (I have two of them myself), or if you want a fan that can be PWM controlled, then I'd recommend Noctua NF-P12 120 x 25mm PWM Fan (P12 PWM).
The GT is about $20 shipped and the Notctua PWM fan is about $27 shipped. Pick your poison.
Ok, now that chosen your replacement fans, now what. Well, typically fans the 25mm variety (most fans in a computer), force air out in all directions and it is not in one directed movement. This causes a lot of air turbulent noise, ESPECIALLY if you have a fan attached to a heatsink or case grill. Typically, you get around this limitation by adding a fan shroud at the cost of extra space. You can 'buy' fan shrouds, but typically I just tear down an old/defunct fan. Now I've already assumed you don't have space for shrouds, so what do you do? Simple... take some tin snips and cut out the Antec 300 honeycomb grills. This will allow air to flow freely into the fan.... but wait! You're probably thinking 'um, I don't want my fingers to get chopped off for one, and it looks ghetto for two. Add your own wire grill back to those fans (Side & Rear) for the most part. Use a WIRE circular grill. (not a laser cut one).
As for your front grills... probably best to leave them as is, unless you want to purchase some magnetic filter grills which can be washed. That should reduce your case / hsf fan noise by a significant amount.
(Like I said, buying the right case is easier / cheaper than modding)
Now, your next big noise maker will either be your PSU or GPU. If your PSU is out of warranty... replace the fan in that with an appropriate Gentle Typhoon fan or Noiseblocker fan. If it is in warranty... and it has a removable grill, remove the grill w/o tearing open 'warranty void if removed sticker'. Corsair PSU's are a good prime example of this.
GPU fans... well, those are a PITA. Thermalshaman... or now closed loop water cooling is an option.. but pricey.
Hope this helps!