Get lost. Wander down alleys. Sit in cafés and watch life pass by. Don't eat where the tourists do, but where you see locals. Just set off down a street and see where it leads. Look around the bends, over the rises. Get away from the crowd. I find that if I meander away from the tourists and tourist sites, away from what is too familiar and comfortable, it's much easier to adapt to the rhythm of a place, and to be more observant.
The more willing you are to travel at a moment's notice, the more opportunities you can access. Day jobs will limit travel, so will mortgages and car payments. Photojournalist Lynsey Addario recently wrote about being 7 months pregnant while on assignment in Gaza. I deeply admire her bravery and commitment to her work, but I imagine many photographers aren't willing to make such compromises. Consider your lifestyle, and how much time you're willing to spend away from home. As for myself, I'm a long-term digital nomad traveling with a suitcase and a backpack and an open mind. Being available and flexible has made a monumental difference to my career.