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The apartment SHIELD puts him in doesn't have a TV. Steve isn't sure if they're trying to separate him from the outside world for his sake or for the world's sake. But Steve Rogers is nothing if not determined and so he makes it his mission to find a way to keep aware of the world around him once they start talking to him about the Avengers Initiative. He soon learns that movies no longer play news reels in front of them, and he's only allowed to go to prescreened movies in the presence of an agent anyways. And if he's being honest with himself, he doesn't really have much interest in wandering around the city. It's too different, and he's still trying to make sense of the world and the people he left behind, much less a world that feels completely different from the one he knew.
One thing he can take comfort in are newspapers. They may look completely different, but they're a familiar thing none the less. So he hordes them, reading through as many archived papers as SHIELD will let him, eagerly absorbing everything they say. At first he reads only the news section, desperately trying to make sense of the changes the world has gone through in the sixty years he'd been under, but soon he finds himself curious about other things. Specifically, sports.
When he first woke up SHIELD tried to ease his entry into the modern world by playing the radio broadcast of a game he'd attended. A Dodgers game. So when an agent brings him a copy of the morning paper, Steve eagerly flips to the section labeled SPORTS. But there's nothing there about the Brooklyn Dodgers. There isn't even a Brooklyn team anymore. There does however seem to be a team called the Los Angeles Dodgers. He turns to ask the SHIELD agent what's going on but he's already gone.
He may be out of his time, but Steve Rogers isn't an idiot. But this, this makes no sense to him. Could they have actually moved the Dodgers across the country to Los Angeles? He spends the whole day stewing over it, infinitely frustrated that he doesn't have any way to figure this out because of the restrictions SHIELD has put on him. In retrospect he might realize that showing real interest in the changed world is a good thing, but right now he just wants to know what the hell happened to the Brooklyn Dodgers.
It turns out that it takes another few months before he actually gets his answer, because that night Commander Fury comes to tell him about Loki and the Avengers Initiative begins in full. And there's no way he's going to ask Tony Stark.
It's not until they're in Brooklyn, examining the wreckage of the Chitauri attack (that didn't stay in Midtown, despite the Avenger's best efforts) that he realizes Ebbets Field is gone and a block of apartment buildings stands where it once was. So he asks (aks Bruce and Clint because even if they're not at each other's throats anymore he still doesn't trust Tony Stark to not roll his eyes). And it turns out they're really gone.
* six months later *
It may have taken a while, but the Avengers have managed to gather themselves into something at least resembling a team. They're all living in Stark Tower (Steve still thinks its ugly, clean energy or not) and Bruce's timebomb description isn't nearly as accurate as it once was. They have PR handlers and the world seems to be slowly accepting the idea that the Avengers are there to protect them. So when the Dodgers new ownership asks one of them to throw out the first pitch on opening day, Coulson asks Steve is he wants to do it.
And Steve says yes. There's some talk of him wearing his Captain America suit but he shuts that down immediately. Instead, he goes out there and throws the first pitch wearing the sixty year old Brooklyn Dodgers cap someone (he suspects Phil Coulson) rescued from his old apartment.