If you're time travelling back, you don't need to time travel day by day or keep anyone from moving before you TT. Traveling back any amount of time counts as one day forward, so even if you went back by 10 years, it would be as going to sleep and playing the game tomorrow.
If you're time travelling forward, that's when the time that you skip counts as regular time and you need to be careful. To add to the thing about villagers pinging, you should also have Beautiful Town enabled and remember that you'll probably get bed hair and some weeds if you skip all that time at once. When you arrive at the date you want to play, you'll also have to talk to your villagers at least once to get the "I haven't seen you in a long time" dialogue.
For example, if you're on the current date but want to go back to December 2014, you can simply set the date back and it will be as though only one day had passed. If you want to go forward to December 2015, that counts as several months passing so you should get someone to ping you and deny their request before skipping in one go.
- - - Post Merge - - -
(By the way, a quick way to get villagers to ping you so you can skip: save and quit your game as your mayor. Set your 3DS clock to the date you want to skip to. Start the game as a new character and instead of going to the town hall, just look at the map once you get it. The villager missing from there will be the one to ping you next, so quit without saving. Set the date back to whatever it was and use your mayor to find that villager and talk to them a few times. If they're not pinging, save, quit, load, and try again. Keep doing that last step until that villager asks you to move. You can then say no and save and quit, then time travel as usual. If nobody is missing from your future map, you can check if anyone is in boxes using that new character. If nobody is, that means that nobody is thinking about moving. I think that then you can just time travel with your mayor without worrying, though if you want to be extra safe, you can save the game as that new character and then load as your mayor on that date.)
