I'm in the middle of a 40-day, 16-stop, 14-country tour of the world.
This has become our process.
Arrive at the airport. Make it through customs. Explain that no, we're not carrying any baggage except these two backpacks. Fight off the taxi drivers, hungry for a taste of American spending habits. (We are always a disappointment.)
Try to figure out if this city runs on buses, trains, or foot-power. Find the quickest way to their most widespread and cheapest form of transportation. This isn't easy - airports hide local transport well, but advertise all kinds of over-priced "expresses" that take you right past the most interesting places and into the areas optimized for spending money. Finding the information we need sometimes means cornering locals, hacking someone's wireless network, or just trying risky things that may or may not work out.
Change any money we have into local currency. This won't be a lot - usually just enough to pay bus drivers or whichever form of transportation requires coinage. The money changers give us weird looks. Sometimes they don't have what we need - Nepal, for instance, runs on a currency even their government won't accept.
Take the local transport to a place we can spend the night. This forces us to figure out local customs, local currency, and local geography, all in the process of getting there. We also have to learn to read their signs, and navigate the peculiarities of their transit system (in Hong Kong, they don't give you change, in New Zealand, they operate on the honor system, in Nepal, every form of public transportation involves being touched intimately by strangers).
For the currency, try to come up with a mental trick for remembering the exchange rates, so we don't get scammed. If their money is a half, or a quarter, or a tenth of ours, then it's easy. Most of the time, it's more difficult. My overall pattern is to figure out the order of magnitude (pesos are roughly 1000 times less valuable than the US dollar), and only then worry about the multiplier (1000 Chilean pesos is worth 2 US dollars). That way if I screw up, I don't screw up quite so bad.
Another trick? Almost every currency uses bills for amounts greater than $1 US, and coins for amounts less than that, no matter what units they count in.
If we've managed to get to a place we can spend the night, then most of our problems are solved. We've reverse-engineered the most essential skills necessary for survival in our new location. We've gotten our feet wet dealing with the local people, and working across whatever kind of language barrier there is. We know how to travel. The city is ours.