Check all of the hoses going to the radiator, upper radiator hose, lower radiator hose, the 2 hoses going to the heater core. The Heater Core is located behind your glove box and very difficult to get to. You have to take apart the dashboard to replace it. Not an easy task if you are not to handy. It could be as simple as an old cracked hose or as difficult as a cracked radiator. More than likely since you don't have Antifreeze or even smell it inside, it's not the heater core. You are going to have to get under the vehicle and look at the hoses. Use a good flashlight and check. If you are pouring Antifreeze and it is coming out right away without even starting the vehicle, then it could be the resevoir itself that is leaking.
Flushing the system is not the best option right now because you don't know where the leak is.
First thing is to find the leak and FIX it. don't get one of those cheap fixes, if it's an old hose, replace it, if it's just a loose hose clamp get a new hose clamp and tighten it up.
Once the leak is fixed then you can go ahead and do a flush. It's not difficult to do a flush. I did one with a buddy in about an hour after I picked up my tahoe.
Below is a link and it shows how my tahoe coolant looked after I flushed it.
http://tahoeforum.com/showthread.php?p=624544#post624544
Locate the drain plug and Drain the radiator.
once it is almost empty, fill up the Coolant resevoir with water from the garden hose and keep the drain plug off and this will get most of coolant out. Wait until it starts coming out clean.
Put your radiator drain plug back on and fill with your choice of coolant flush cleaner and water from the hose. Follow the directions on the bottle, more than likeley it will say to run the vehicle for about 10 - 15 min. Put the heat on MAX so the heater core is getting the flush cleaner running through it as well.
Turn the vehicle off and drain once again.
Disconnect the hoses from the water pump, not the firewall. The plastic fittings on the firewall tend to break very easily. Run the water hose through the heater core hoses to flush that out completely and once again wait until the water runs clear. While you are there you may want to replace your thermostat.
Once everything has been drained and water has been running clear through everything, you are now ready to fill up with Dexcool. This is the orange coolant, NOT green. You have your choice of getting the 50/50 pre-mixed, or mixing yourself. Either way fill it up and turn the vehicle on again and put the heat on MAX again and leave the radiator cap off so the cooling system and burp itself for a little while. This can take a little bit but shouldn't be too long.
Cap the resevoir and you should be good to go!