You don't need to buy all of the expansions immediately, but your game must be up to date to play on the servers. You can just copy the game files from someone else if the download size is a problem. Ask in the forums here and on MyBroadband to see if someone is close to you.
If you don't have all of the expansions, then you won't be able to access the zones for those expansions. You also won't be able to play the races and classes that came with that expansion and will be limited on your max level.
If you buy the battlechest, you will have access to 10 races, most of the Eastern Kingdoms and Kalimdor zones as well as Outland, and have a max level of 70. You can add WotLK at a later stage, which gives you DK's and the Northrend zones, as well as a max level of 80. When you add Cataclysm, you can get to level 85 and play all of the zones. You also get Worgen and Goblin races.
