I wouldn't want to store btc anywhere online.
Do you think your own computer is more safe than a hosted solutions with 24/7 ops peoples and automated backups? I don't ask this question facetiously. Your own machine may very well be more trust-worthy, but remember that one non-backed up hard drive crash could wipe out your wallet.
What's meant by the "blockchain loading up"?
The blockchain is a record of every single bitcoin transaction that has ever happened. In order for the system to know if a transaction is a legit, it has to go back through the blockchain and make sure the sending wallet was once credited with at least the amount being sent, and the wallet before that, and the wallet before that. Basically, If you're giving me $20, I need to know that someone else, at some point in the past, gave you AT LEAST $20 (or more) AND that you didn't spend that $20. Naturally, the only way I would know that is if I knew that someone gave that person at least $20 (or more).
Sooooo.... yeah, the blockchain must be "loaded up" so that client software can verify past transactions. It only has to happen once, but could take a LONG (as in "days") time.