The parking lot opens about an hour before Magic Kingdom opens to the public.
Since the first arriving guests park quite near the tram station (you walk down an underpass to get there), the trams don't start running immediately when the parking lots open.
The monorails and ferry may be running that early but sometimes (probably on extra magic hour mornings) don't begin until 20 minutes before Magic Kingdom opens.
I'm not exactly sure about the ticket booths, they should be open when the parking lot opens, but you can also buy tickets "at the other end of the monorail ride", that is, just in front of Magic Kingdom.
They also have ticket machines set up at some of the parks so if you know what you want, they might be faster. They are probably also operational before the actual manned ones are.
You can save money by buying your tickets in advance. There are a lot of sites discussed here. Even if you buy from Disney online, you will save money over getting the tickets at the park.
If you are staying at a resort, you can purchase your tickets at the guest services counter before you leave for the park for the day. You could also purchase your tickets at a Disney Store (if you have one near you) prior to leaving home.
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