Rio Gallegos is a city in the south of Santa Cruz province, Argentina.
North towards El Calafate/Trelew
The Ruta 3 goes right into Rio Gallegos, so you just gotta go to where it leaves the city, and there are some decent pull-outs where you can stick out your thumb. But if you're having trouble there, or someone offers to take you, about 20 or 30 km down the road is a police checkpoint right before it splits to with the Ruta 5 to the west and Ruta 3 to the north.
South towards Punta Arenas/Ushuaia
Avenida San Martín turns into Ruta 3 if you just walk far enough (a few kilometers from the center). Eventually, you'll get to a spot that'll work for you. If you want a truck, though, (which is especially useful if you're going to Ushuaia), it might be better to try one of the two petrol stations (YPF and Petrobras) near the bus terminal.