Your first step needs to be to make a list of everything you want to continue watching and then figure out how you will keep doing that.
In general your PC(s) are better at local video playback. They also make great DVRs. Your streamer boxes like Apple TV are best at just that - pulling in and navigating web content.
-you can install an antenna and ATSC tuner card in your PC to watch and record ABC, CBS, NBC, PBS, FOX, The CW, and more.
-many of those programs (and some from cable channels) can be found and purchased/watched on Hulu+, Amazon, and/or iTunes.
-Netflix disc rental and your library can fill in the gaps. You might be very surprised what you can get from the library.
If you like sports you are going to be a tight spot, especially if you follow your local teams. The NFL will be the easiest to continue following. Those games are all on major networks. Even the ones on ESPN and NFL Network here in Pittsburgh are simulcasted on the local ABC and CBS affiliates when the Steelers are playing.
NHL, MLB, and NBA are a bit tougher. Each has some games on broadcast networks (NBC, FOX, and ABC respectively). But they also broadcast some national games on cable (NBC Sports, TBS/ESPN, and TNT/ESPN). And finally the bulk of your team's games will be found on your local cable affiliate.
Each of those leagues offers a web service, but with a few caveats. Most importantly if your IP address is in the local geographic area as your team you are out of luck. If you subscribe to NHL Game Center, and the Penguins are on Root (which is technically available to you in the South Hills), you will not receive the game as they are trying to protect the local affiliate (in this case Root Sports) who paid a lot of money to offer you the game.
Secondly even if you follow an out of market team (say the Rangers) you will still have a lot of games blacked out. When they are on any nationally available network (NBC, NBC Sports, or random games NHL Network decides to pick up) you will not see it. The playoffs are also all on nationally now. I'm pretty active on a Penguins message board and we determined last year that around 30 games were either on NBC, NBC Sports, or were selected by the NHL Network on a random Tuesday night. That means anyone who paid for Game Center with the hope to watch the Penguins lost out on about 40% of the games.