Mac mini won't connect to internet despite displaying wifi connection

Question

My Mac Mini says it is connected to the internet, but when I attempt to load a website it just continuously says the webpage is loading for a few minutes until eventually saying "This webpage is not availabe" (Chrome).
The internet works fine on all of the other devices in the apartment (iMac, Macbook Pro x 2, iPhones, Windows laptop, PS3, etc.).
I have tried to find the answer elsewhere on the internet but nothing has seemed to work.
When I run diagnostics, on the lefthand side the ISP, Internet, and Server dots go from green to red every few seconds. When they turn red they each say "Failed" underneath them.
I have no idea what the issue could be. The Mac Mini is running OS X Mavericks and its software is up to date.
Another point to mention is that, even if the wi-fi is working on the other devices, if I reset the wi-fi it occassionally works on the Mac Mini for a couple of minutes or so.

Answer 1

First off you are making the common mistake of confusing the WiFi LAN (Local Area Network) with the internet (a.k.a. WAN or Wide Area Network). Your LAN is the gateway to the internet. It is entirely possible to have a solid connection to the LAN and all the devices on the LAN without having an internet connection.
When you say "..reset the wifi.." did you reset everything or just the router?
The correct procedure would be to
begin with the devices on the LAN (WiFi network) and power each of them down,
then power down the router,
if there is a separate modem end by powering it down.
Give everything a good 30 seconds or so
power up the modem and wait for it to connect and stabalize (all the lights quit flashing.)
If there is a separate router then power it up and wait for it to stabalize,
next power up the mini and wait for it to connect then test to see if it is connecting to the internet.
Finally, one at a time power up each of the other network devices waiting for each to connect before going to the next.
It is important that you reset everything and the order in which you reset the various network devices is CRITICAL.