While this will definitely lead to better and healthier plants in the long run, some gardeners get discouraged by problems with weeds or pests. Although chemical 'fixes' might work on a weed or pest today, chances are very likely that those clever weeds and pests will soon develop a resistance to the chemicals making them useless and will make them even more difficult to destroy in the future. As far as the weeds, nothing will get rid of them better than old-fashioned weeding. Start pulling weeds as early as possible and do a little weeding each time you are in your garden. If you get them early they're much less likely to overrun the garden.
Pests are usually a much bigger problem for most people.
There are so many pests that there's no way I could get into the specific ones here. Check out this site to learn about BUGS - the good, the bad, and the ugly in your garden. The best way is to let nature take its course. By that I mean that virtually all garden pests have natural enemies, so let them do the work for you. It is very important to remember that not all bugs are bad - in fact many are good.