If you're here then it's probably because you just got the game, and the thin instuction book, vague in game instructions, and lack of online guides left you confused.
How time flows
In game, you can spend as much time as you want picking daisies and prancing in general. However, in order for the date to change you need to use that little fast forward feature the game showed you. During the day, animals move around and stuff in general fits into place, but won't update until you send it into the next day. No matter how long you watch the hungery rabbit stand next to some grass nothing is going to change unless you fast forward.
How to feed your animals
Your animals will move to whatever it is they eat, they just have to be able to reach it. So if a hungery marmot is facing a pile of seeds right next to it, then you're good. As stated above, you need to fast forward a day otherwise the game won't update and nothing will happen.
There isn't much you can do about fences other then get a feel for how far your characeter throws it. You always have to use the hand tool to pick up fences though.