After you leave the previous city you'll enter a forest. You'll see a man standing near a cut-able bush. Talk to him and he'll tell you that his boss's Farfetch'd got away.

After him you will see the Farfetch'd by itself. Chase it around until it reaches the man by the cut-able bush. Talk to the new man and he will give you Cut.

Now Cut that cut-able bush to get passed the only one Blocking your path. A new feature in Gold/Silver is that you can just go up to the bush, press A, and it will get cut down if you have a Pokemon with Cut in your lineup. The same goes for Boulders, and that Tree that will come up later.

Inside the Ilex (Ubame) Forest Maze is a guy who will give you TM02, Headbutt. Headbutt lets you shake the little trees that are everywhere in the game, sometimes revealing sleeping Pokemon.

I found an Exeggcute and #190 Aipom, that little monkey with a hand-like appendage on the end of its tail. Beedrill was also found in Silver, Butterfree in Gold. Heracross is also up in the trees, but is very rare.

After cutting the bush that is right before the shrine dedicated to the Forest's protector (aka Celebi), you'll wind through the Forest path until you come to a connecting gate. Inside is a girl who will give you TM12, Sweet Scent. It will attract Pokemon to you outside of battle.

Once you exit this place, you'll come to Route 34. Amongst the trainers is a Daycare/Breeding Center.