I have never really time traveled because I am scared, but I would like to get things done quicker. If I just time travel day by day, are there any cons to it? Also, lets say I time travel day by day for a month's time then go back to the actual date, will anything bad happen then? Will my villagers leave without saying anything to me first?
If there is a thread about the pros and cons of time travel, a link would rock.
Well, it'll simulate getting on once a day, basically. If you water your flowers and perform a day's tasks normally, nothing should change. You're just skipping a day instead of waiting 23 hours.
As long as you go day by day, you shouldn't have a problem. Just make sure to try to keep in touch with your villagers so they will ping you when they are thinking about leaving. If you do this, like PaperLuigi said, it won't make a difference.
And yes you can skip back to the current date in one leap, and it will only count as TT'ing one day. Just make sure you don't do the reverse and go forward in one leap, that's where you'll have villagers moving away and flowers dying.
Going back in time won't be a big deal because going back in time any amount of time is the same as going forward one day. When you travel back, it'll be like a day passed in your game.
Going forward day by day can be dangerous or safe depending on what you're doing. If you get distracted, it's easy to accidentally move out villagers. Just make sure you're keeping track of your villagers so you'll know if they're planning on moving. Other than that, the only side effects are things that come with passing time like weeds and wilted flowers but you can stop that by setting the Beautiful Town ordinance.
I think you need to be very careful about why you are time traveling. If you have simply done all there is to do in a day and want to load a new day, I don't think you'll run into any trouble at all. And as you've heard, a big jump back to the present is no issue. I think people run into trouble when they start having other goals and use time travel to meet those goals - things like breeding flowers, looking for new items in the shops, looking for Redd or Gracie checks. When you have those other goals, you start to move the clock without taking care of your villagers, and then you run into problems of missing a ping when someone's going to move out. So that's when you need to be very careful to know that you've looked at everyone and remembered how long it's been since you last had a ping.