I realize this may be a gender thing as I have never hesitated asking directions and it's saved me tons of money travelling around various cities. I figured out my way around using the subways and buses of New York. I even used the metro in Paris (which is really extensive) to get around, took buses a few months ago in London and Amsterdam too. Yesterday, I returned home from LA where I got around the city using the metro and buses. I spent a lot of time on foot in these cities too to get a real feel of the places I've visited.

The first day in LA I took a shuttle from the airport. Spent two horrendous hours with 8 others and a driver that obviously was either new or inept I thought I'd use cabs. I did the first day and found the drivers asking me for directions. Urg that was frustrating. The following day I got out my map spoke to one person and found the metro and buses were really easy to fiquire out. I ended up taking the metro and a bus to LAX which cost me aprox 8 dollars compared to the 45-60 it would have cost to take a cab.
Still saying that, it's also adviseable to be safe, as bringing attention to oneself as a tourist in a strange city may bring out bad characters who see you as a easy mark. Knowing my destination beforehand and the main streets helped as bringing out my map in some neighborhoods would make me an easy target I'm thinking. The only incident I encountered was when a homeless guy tried to snatch my coffee that was sitting beside me when I was sitting on a bench. I snatched it before he could grab it. Later walking by the same place I purchased a coffee and cookies and just handed it to him when I walked by.
Overall, travelling around via buses, metro and lots of walking gives one a great feel of the city you're visiting...when in Rome kinda thing. Admittedly, I haven't rented cars when visiting cities as driving on unfamiliar streets would scare the heck out of me. Although I wouldn't hesitate to ask for directions!