Bulbasaur: Defeat every trainer in Viridian Forest and talk to a guy in Viridian Forest.
Charmander: Defeat every trainer in Route 24 and talk to a NPC in Route 24.
Squirtle: NPC in Vermilion City, requires beating Lt. Surge.
Second fossil: Talk to the same guy that gave you the first one once you beat the League.
Moltres: Victory Road 2F
Mew: Use Surf in the Vermillion Harbor. Use strength in the truck. You need to beat the Champion once to do this.
Mewtwo: Same location as Fire Red/Leaf Green. You don’t need Rock Smash for it, there is a path without it.
Check any Pokémon location
Make sure Pokédex Mode is set to Easy in the Options menu
Look for the Pokémon that you want to find in the Pokédex
Press A in it’s entry until you reach the Area screen
Items
DexNav: Shows wild pokémon in your current location. Trainer School in Viridian City
Appeal Sensor: Lets you see your Pokémon conditions (cool, beauty…). Given by a girl in the Contest Building in Vermilion City
Watering Can: Allows you to water berries. NPC in Route 7.
Light Ball: To improve your Pikachu stats. Buy in Celadont Department Store, 4 Floor.
Upgrade, Metal Coat, King’s Rock: To evolve certain Pokémon. Buy in Celadont Department Store, 4 Floor.
Technical Machines: Allows you to buy every TM. They get unlocked as you get more badges. Top of Celadon Dept Store, there are machines to buy them.
Linking Cord: Evolves Pokémon without trading. Key Item. Defeat your rival in Silph Co.
Shiny Charm: Can’t be obtained yet.
Rock Smash: Can’t be obtained yet. You can get to Mewtwo without it.
Misc things
Stats Judge: Allows you to see IVs in the summary screen. NPC inside the Pokémon Lab of Cinnabar Island
NPC to check Hidden Power type: NPC inside a house in Lavender Town.
Move reminder: NPC in Cerulean City, costs 5k
Power Items: Allows you to win 2 times the EVs of the selected stat. NPC before entering the Elite 4. You must be champion first.
Move Tutors: NPCs of Move Tutors removed. You can obtain those moves with the Move reminder.
Trainer Card Stars
Complete the Pokémon League
Catch every Kanto Pokémon
Obtain every in-game achievement
Obtain every Gen 3 Pokémon (requires trading)
Achievements
Since version 1.8.0, the game has as in-game menu called Chievos. Those are purely made to track progress and small achievements, they don’t give any type of notification or reward when completed.