The blockchain is the full record of all Bitcoin transactions. Without it, we wouldn't know how much each address is able to spend.
There are some clients (called lightweight clients) that rely on their peers' blockchains while others like Bitcoin Core (sometimes called a full client/node) store the blockchain themselves. If you choose a full client, you should expect to download and store the blockchain (~21 GB currently). You can speed up the initial sync by download bootstrap.dat from a torrent, as instructed here.
What you're describing, though, is not cold storage! Cold storage means that the keys are only accessible via offline means, like a piece of paper, or a computer that is never connected to the Internet.