Not sure which game to buy? Here are some of the pros and cons of each game in the series.
If a feature is listed under a specific game, assume it’s a console or game-specific feature and was not continued into the next game, i.e. traveling to the island with GBA link cable, unless specified.

GameCube:

• Biggest town (15 neighbors)
• Four individual houses for players in one shared area
• Acre system
• The most amount of holidays/celebrations out of all games
• Neighbors actually participate in holidays (not just walking around/talking)
• Tom Nook runs a raffle on the last day of the month
• Ask neighbors for jobs
• Cooper and Booker have a police station
• Train station
• Igloos/tents neighbors set up during specific months
• Island you can visit if you have a GBA link cable
• Original town charm
• Most varied/creative conversational content
• Can take a train between games saved on memory cards

Wild World:

• Smallest town (8 neighbors)
• Updated graphics
• One house shared by four players
• No more acres
• Holidays are not related to the real world/barely any at all
• Don’t have to mail fossils away to have them checked
• Able Sisters sell clothing instead of Nook
• Accessories/hairstyles added
• More bugs/fish/fossils
• Players can now learn emotions from Dr. Shrunk
• Wi-Fi
• Portable 

City Folk:

• Larger town (10 neighbors)
• Upgraded graphics
• Four individual houses for players scattered around town
• Decent amount of holidays are back
• DLC items from Nintendo
• Neighbors will not have something new to say unless you leave the area
• More clothing/hair/accessories 
• Player can go to the city
• Auction House lets you sell things to other players
• Grass disappears with wear
• Essentially Wild World for the Wii with upgrades

So what do all the games have in common?

• Nook’s store upgrades
• Nook has Stalk Market
• Player must pay off debt to upgrade house
• Neighbors ask players to do errands 
• K.K. Slider plays music on Saturdays
• Player can complete collection and donate fish/bugs/fossils to the museum
• Player can make custom clothing designs
• Same fruits in every game
• General look of town stays about the same
• Towns all have individual layouts
• Many of the same NPCs in each game
• All games begin with preconceived story (Train/Cab, work for Nook… etc.)