The Stargate network is basicaly a giant super computer network, with each gate able to communicate with any other gate in the network, given that the network safeties aren't tripped(passing through a sun being one of these safeties).
The gates obviously can compute millions of times faster than anything on earth, and the only purpose that the SGC computer system serves is to dial the gate, and read the information it picks up from the gate. The gate itself does everything else(case in point, the earth Stargate can still function without a proper DHD).
When one gate dials another it requires that the reciving gate be "turned on", much in the same way that you need someone to pick up the other end of the phone when you make a call.
Normally the gates sit in a stand-by state, minimal power with the inner track locked(Tantilus). But when you dial that gate, it powers up to recive the wormhole and any travelers that might be comming through. When power is introduced, the inner track unlocks and spins, incomming or out going.
The gates work much like a telephone, but far faster and over much greater distances. Your phone, when dialed, stores the number in itself untill you reach a certain prerequisit(example, 555-1234 will send, unless you place a 1 in front of it, then the prerequisit changes and it waits for you to reach the new point) and then sends the entire sequence in one burst over the phone lines to contact the reciving phone, this of course has to go through a routing station to be directed to the right phone.
With the gates however, the transmiter and routing station are all contained within each ring. You dial, and when you finish, in less time than it takes you to blink the "number" is sent to the "routing station", and a signal is sent to the reviving gate to open a wormhole. The cheverons lighting up on the reciving gate are just an indicator of the reciving gate powering up and serve no real purpose other than to let people near the gate know that there is an incomming wormhole, and that they might want to step back if they are right next to the gate(another saftey feature to protect those who are too close from getting caught in the "blast" of the gate opening).
It takes a while longer than the series shows to establich a wormhole. If you think about it, the closest system to ours takes light far longer than a year to reach us from that system, so it would take the Stargate system a little while to send a wormhole as well.
Just remember, these gates were built by a species far more advanced than humans with a far better understanding of these things. They flew an entire city out of our galaxy for crying out loud.
Anyway, it's just an idea on how the gate system works, even I don't know how they actually work. But what I typed sounds like a really good way on how they could work.