Are really all transactions (the whole blockchain) stored on every client?
Everyone running a bitcoin Core node must download the entire ledger of transactions listed on the blockchain... But the vast majority of Bitcoin wallets are simply a place that stores a user's private key and allows users to interact with the network. When someone sends a transaction from their wallet, it gets broadcast to the network and eventually to Bitcoin miners. 

Once a miner finds the next block (that includes your transaction), the solution gets broadcast to the network and those that are running the full Bitcoin Core software incorporate that block into the blockchain.

So although not everyone needs to download the full bitcoin blockchain, there are still a very large number of people who do, and by doing so, secure the network.