This small hint of Route 4 features a Pokémon Center for you to heal up in. Talk to the portly man inside the Center for a unique opportunity. He is willing to sell you a Magikarp for P500. Magikarp and Gyarados - Recommended Magikarp sucks. It's stats are poor, with the exception of its 90 base Speed. It might be worth your money to get it this early, because for a while it only knows the move Splash, which does absolutely nothing. Magikarp is a pure Water-type. You get it at level 5 and it is not until level 15 that it learns Tackle, which it does not have the Attack stat to get much use out of. However, if you raise your Magikarp all the way to level 20, it evolves into the almighty Gyarados, a fearsome (not-actually-a-)dragon with an Attack stat to match and great Special too. With this great power comes the responsibility of having a Flying subtype, which brings a 4x weakness to Electricity to the table; however, Gyarados can make good use of everything from Blizzard to Thunderbolt to Hydro Pump to Hyper Beam. It even learns the latter two by level up! Competitively, it is checked by it's Electric weakness, but in single player, a Gyarados is a scarcely stoppable destructive force that will make your game much easier.
After you've made your decision, exit the center and enter Mt. Moon.
There is little to do in Pewter City beside healing up at the Pokémon Center. The Poké Mart does contain one new item.
Poké Balls - P200, it's always a good idea to stock up on these.
Antidotes - P150
Awakenings - P250, these will heal the Sleep status ailment. It would not be a bad idea to pick up one or two.
Burn Heals - P250
Parlyz Heals - P200
Potions - P300
Escape Ropes - P550, these items can be used outside of battle and will return you to the doorstep of the last Pokémon Center you used. Their best use is when you are deep in a cave with low health. Note that they are frequently found inside such caves for free.
If you want to brush up on your history, head into the Pewter Museum of Science, to the north; however, this is not required for anything and costs P50. When you're finished exploring. Enter the gym.
The first thing to do upon entering any town is to head to the Pokémon Center, where the nurse will heal your Pokémon for free. This restores their HP, PP and status. In the northeast corner of the Center is a PC. Not only can this one store items, but it can store Pokémon as well, up to 240 of them. If you have access to another Game Boy with Red or Blue Version and a link cable, you can battle or trade with them by talking to the attendant on the right. After finishing up at the Center. Head northeast to the Pokémon Mart. Upon entering, the clerk will identify you as being from Pallet Town, and give you Oak's Parcel, which you need to deliver to him. You can't buy anything until you do, so exit. One more thing you can do here is walk clockwise around the clump of trees in the northwest part of town. Examining the cuttable tree with A will yield a hidden Potion. Next, head back to Pallet Town.