Any trusted Broadway/Concert Ticket Agencies? (New York)

Hello All,
I have another question ... is there a trusted site /ticket agent for buying theatre shows here in NYC? There are so many on the net and the prices vary quite substantailly, anywhere from $60 to $275 per ticket for some of the current regular shows in town.
How about turning up at the theatre on the day for spare seats/ or returns ?
Do they do that here?
In London, most theatres, you can show up at the box office (the ticket counters) in the actual theatre, they usually reserve approx Protected content a day for release on that day for that day's performance. The tickets are usually much cheaper than buying them in advance. They may not be the best seats but I have over many years in London have got front / or stall row seats for a bargain. It is on a first come first serve basis. I'm just wondering if NY has the same system/policy here?
Sorry..that's about three questions here. Any wisdom on this is appreciated.
Thank you.