Time traveling, simply put, is using the "reset clock" feature in the options menu to change the time and date. This is useful for getting things to happen immediately (e.g. getting fruit to reappear after you've picked it all, getting mail sooner, and so on). In order to do this you will need to pick up the phone in your attic before you wake your character up and click 'Reset Clock'.
Time travel will not bring characters back to your town, nor will it make a special person reappear in your town on the same day and time (unless it is a scheduled person, i.e. Joan or Redd). Also, if you miss a balloon or U.F.O. in the sky, time traveling back to the same point you missed it won't bring the item back, either. These are just a few problems that can't be solved with time travel.
Weeds do come up when time traveling to the future and some flowers may wither but traveling to the past is easy to get something from Nook's Store that you missed and so on. Be warned: if you time travel too far into the future you will find your town covered in weeds and the residents will be unhappy. Tom Nook's turnip prices will never go above 96 bells, and Joan will only sell them over 96 bells so you never make a profit from white turnips. Your favorite characters may be more likely to move away.
Time travel is almost never a good option. Just be patient.
However, If you need to time travel but only need to go forward or backwards a small distance (e.g. 1 week), do not do it all at once. Traveling forwards or backwards one day at a time enables you to find the 3 weeds that would pop up daily instead of the enduring task of picking out possibly hundreds. You can also look in the shops without missing anything. If you are time traveling to grow a tree, do not time travel four days in advance or the tree will rot. As advised, only travel 1 day at a time. Remember that it takes 6 days for a tree to fully grow. When you want to go back to the original follow the same rule or you will get even more weeds still.
To fix these problems, synchronise your AC:WW clock with your Nintendo DS clock. They should go away.