The current Pokémon Champ, Red, is from Pallet Town, as is Blue and his grandfather, Professor Oak. Red is off on a trip, so you won't be able to challenge him just yet. You can visit Professor Oak in his lab, if you'd like.
House of Blue's
Blue, Viridian City's Gym Leader, is from Pallet Town, and his sister still lives there. If you visit Blue's House (in the upper right corner) from 3pm to 4pm, Blue's sister will groom one of your Pokémon. Pokémon love being taken care of and prettied up, and if you groom them often, they will become more attached to you. You have to visit the house at that specific time, however, or Blue's sister won't work on your Pokémon. You can have only one Pokémon groomed per day.
The Professor is in
Visit Professor Oak's lab to touch base with the Pokémon professional. Professor Elm has emailed Oak about your arrival (you can read the letter on his computer), and he's eager to speak with you again. You can show him your Pokédex and get his evaluation of it on the spot. Professor Oak is interested to learn that you are collecting Kanto Badges. He asks you to return once you've collected all eight of the badges.
The home of a champ
You can also visit the home of Red (in the upper and speak to his family. Red isn't around, but a quick check of his room reveals that he upgraded to a Nintendo 64.