The woman in the building between Ilex Forest and Route 34 will give you TM12 Taunt. This move forces the opponent to only use physical attacks.
Now you can head north, battling trainers and catching wild pokémon. You will possibly encounter Abra, which is a psychic pokémon that is difficult to catch. Using Taunt will prevent it from running away from battle, although if you have a Quick Ball, you can use it on the first turn of battle, and Abra's catch rate is high enough that it's a guaranteed catch!
On the way north, your best friend will introduce you to the Day Care Center. This is where you can drop off one or two of your pokémon to let the Day Care Man and Day Care Lady raise them for you. The down side to this is that when a pokémon in the Day Care Center levels up, it will automatically replace one of its current moves with a new move if it reaches a level where it learns a new move.
Also, the Day Care is important because if you leave two pokémon there who are compatible, then they might make an egg! If this happens, the Day Care Man will call you (probably) and he'll be facing the water instead of looking south. While waiting for an egg, you have to walk around. After a certain number of steps (depending on the pokémon compatibility), there is a chance that an egg will be created.
In the northeast area you will find TM63 Embargo. This move prevents pokémon from using items. It's not that useful for NPC trainer pokémon or wild pokémon, but it could be helpful when battling against actual human trainers, who are much more likely to use items on their pokémon than their NPC counterparts.
Now you can head into Goldenrod City. It's pretty much the biggest city in Johto, and there is so much to do here.
First, there is the Department Store. Here, you can buy all sorts of items. Also, on Sunday, there will be a woman at the 5th floor counter who will either give you TM21 Frustration or TM27 Return depending on your pokémon's happiness.
Also on the fifth floor is a guy who will trade you a Machop if you trade him a Drowzee. Additionally, there is a soda machine on the top floor, which has some heal-up beverages that cost less per HP gained than the different potions. Unfortunately it takes longer to get the same amount of soda machine items as regular potions.
Finally, there are two women at the counter on the top floor. If you pay the bottom woman $300, you will get to do the raffle drawing for either the number 1, number 2, or number 3 prize. The prizes depend on what day of the week it is. The Number 1 prize is always a TM, the number 2 is always a type of pokéball, and the third is always a berry.
The elevator of the Department Store can go to the basement, where if you talk to some Machokes, you can find some items.
Answer the girl's questions correctly to receive a radio card for your pokégear.
A woman at the counter will check the Trainer IDs of all of your pokémon (including those currently in PC Boxes) and if the digits match the random number that she generates, you will get a prize. You can get a partial match if the last number, last two numbers, last three numbers, or last four numbers of the trainer ID matches the one that she generates. The prize for an exact match is a Master Ball!!
Go upstairs to meet Buena. If you listen to one of the radio stations on your pokégear, you will hear Buena's Password show if the hour is 2, 5, 8, or 11. After she says the password, go meet with her in person here at the radio tower and pick the password from a list of three choices. If you get it right, she adds a point to your Blue Card (which she will give you if you don't have one). After your card accumulates enough points, Buena's assistant will give you a prize as you leave.
You will no doubt notice that the third floor of the radio tower is blocked off by a policeman, who says that the director has been behaving strangely lately. Hmm...
East of the department store is the bicycle shop. Go here and you will get to borrow one for free. It makes traveling much faster! Be aware, though, that if you go through grass while riding a bicycle, you will encounter more pokémon than usual because the bicycle noise attracts them.