What type of router do i need?

DO NOT SKIP THE LENGTH THESE ARE ALL THE THINGS TO ANSWER YOUR QUESTION!!

No 'router' will provide 'really good internet'. A ROUTER connects the " Gaming Desktop, a normal desktop, 3 laptops,1 ps3" together to SHARE whatever 'INTERNET' you purchased from your ISP. I would suggest with this many devices (you should also think about your iPods, iPhones, Android phones, etc., that ALSO will want to use the Wifi) that you have a higher end Internet service, which would mean most likely CableTV Internet, at least the 75Mb/s package. Time Warner been adverting 300Mbps, I was already doubled to 100Mbps.

Now to make all these devices play 'nice' that the PS3 isn't lagging out Mom's Laptop trying to watch Vampire Diaries on Netflix, etc. you will need to implement across the network QoS and on EVERY DEVICE trying to connect to the Internet. QoS is Quality of Service, this provides a 'table' that says who get's priority service OVER other services. As the example Mom won't allow streaming videos to lag EVER, so Video gets #1 priority, but checking Facebook shouldn't keep Timmy's PS3 lagging no matter how many 'Farmville and Spades' Grandma on the desktop likes to play at the same time. So PS3 Gaming would be #2 priority, etc.

That said, the next issue is the signal. Now the 'current' standard is 802.11ac, the older standard is 802.11n that the PS3 is set for, and depends on all the other devices. As noted here http://www.extremetech.com/computing/160837-what-is-802-11ac-and-how-much-faster-than-802-11n-is-it . What this means is more antennas were added, better 'chips' inside so MULTIPLE devices can be sending AND receiving rather than every device having to TAKE TURNS to send OR receive, longer range, faster speed signal between devices and router, and so on.

BUT.. you still need to 'map things out' and 'plan ahead', taking into consideration the basics: Distances, Objects, Interference, Device capability.

Distance: Simply put there is only so many feet before (like your cell phone) you 'drop signal', if you were standing outside with nothing between the device (laptop, PS3, etc.) and staring right at the router it is normally about 50FEET away. As noted here, this can be better or worse depending on the maker (http://www.trustedreviews.com/opinions/802-11ac-vs-802-11n-what-s-the-difference "Linksys EA6500 which hit speeds of 30.2MBps (241.6Mbit) when connecting to a device just two metres away, but still performed at 22.7MBps (181.6Mbit) when 13 metres away with two solid walls in the way. By contrast Linksys’ own EA4500 (identical except being limited to 802.11n) managed 10.6MBps (84.8Mbit) dropping to 2.31MBps (18.48Mbit) under the same conditions.")

Objects: As noted above the more 'objects' and the thicker these objects are, the less signal you will get. So trying to put the router in the 'Game Room' with the nice BRICK FIREPLACE and wonder why you get signal in the kitchen across from it but right behind it in the spare bedroom you get no signal, the SOLID THICK brick wall is blocking the signal. Think about the furniture, walls, PIPES (oh yeah that is where the water STEEL pipes comes down through the spare bathroom!), walls, doors, etc. between ROUTER and where your standing, each thing degrades the signal more and more and more till you get no signal.

Interference: Now if you have other high end electronics, you can cause, like radio stations, interference with each other. So you want to make sure the PS3 isn't sitting on top of those 5 Foot tall Kenwood speakers because the speaker is giving off a large Electro Magnetic field and will interfere with the PS3 able to get / talk 'right' on the Wifi.

Device Capability: So commonly some one complains "why does my brand new iBook allow me to walk from one end of the house to the other, but Bob's Dell laptop can't?". Simply put; depends what the iBook has as compared to the Dell for hardware (wifi card) and software (which Windows / Mac OS). As I said a few times the PS3 is OLD, so it will be on a LOWER capability, and thus will not be 'as fast' nor reach 'as far' as the brand new iBook, etc. which uses the latest 'tech.

Considering the size your stating, most likely it be best to get your FIOS/CableTV mounted in the 'center' of the house as best as possible. Normally the ISP provides a Modem/Router already, but it WILL NOT cover the entire 2,300+ square feet. SO you may want to SPEAK to the SPECIALIZED SERVICES / Business Class group that either recommend the Router that is most 'compatible' with their equipment (some ISPs do limit them, some are very forgiving) OR will have a full package to accommodate "VIP/Business Class needs" that would include all the costs for installation, hardware, tech site services when there is problems, onsite testing, site survey, etc.

IF they have 'comparable' then you need to buy one that would work best with your service here is a couple articles on the 'best' of 2014 http://www.cnet.com/topics/networking/best-networking-devices/802-11ac/
http://www.pcpro.co.uk/features/389392/the-7-best-802-11ac-routers-2014-whats-the-best-802-11ac-wireless-router

Again a ADSL won't work on a CableTV ISP service, as much as a CableRouter work on FIOS. All depends on your ISP you decide on / available to you. That said, then you would need to invest in 'EXTENDERS'. Normally one has a 'map' of their home and figures out where to install the Internet and then best 'guess' on the range (making a circle and taking in the factors I pointed out above) from that point then they map out guesstimate of more 'circle' ranges to extend (overlap from one Wifi circle of coverage to the 'next' circle of coverage) to the areas the person wants. Be mindful, any OUTSIDE signals (out on the porch, etc.) requires specialized WATERPROOFED and CERTIFIED FOR EXTERNAL USE extenders. These are different, but perform the same function. Once you 'guessed' your number of Extenders, then the 'proof' in the pudding is actually metering / monitoring the installed Internet and router (this is covered as I said by some VIP/Business class services where the tech walks around with the laptop in hand, watching the signal and 'maps' out where it drops). Simple use of your laptop and say the free Xirrus Wifi Inspector (http://www.xirrus.com/Products/Network-Management-and-Software/Network-Management/Wi-Fi-Inspector) to validate your assumptions before buying the number of Extenders (ohh dang I forgot about that steel Gun Safe! no wonder the signal drops!).

Now you can as well shop around for a 802.11ac Extender (again just google like I did or read this article http://www.engadget.com/2014/01/06/netgear-802-11ac-wifi-range-extenders/ I personally recommend the Netgear 'Wall-plug' type design as the most unobtrusive and simple plug and setup method to doing this). DO NOT BE FOOLED by stuff like the Amped Wireless' REA20 range extender claiming to cover 10,000sq feet!. As noted in this real world review http://www.overclock.net/products/amped-wireless-high-power-700mw-dual-band-ac-wi-fi-range-extender-rea20/reviews/6441 "Amped Wireless has employed some tricky math in their advertising claims. They state that the product will offer 10,000 square feet of extended wireless/wired coverage. This is generally true, but doing some simple math we find that this only accounts to about 56 feet in any direction, less if there are walls. This is still a great extension, but it makes the 10,000sq ft seem a bit inflated. "

So again, that means if you were 50 feet away in a straight line, it would drop with walls, etc. in the way, just as the 'normal' range of the other router's I pointed to above.