I agree with Transisto and Shuai. Bitcoin needs time and there are a number of different issues that need improving before it will hit mainstream. What I personally think as the most important single issue, is marketing. Bitcoin itself, the tools people use in the Bitcoin economy and the merchants and organizations who accept Bitcoin need a lot more marketing. We need people to know about all of this. Everyone can help with this, just talk to people about it.
We have a technically superior payment system to anything out there. The tools for using it have been improving rapidly. There are already a lot of wallets that bypass the problem of a large blockchain, even for mobile phones! For desktops we have at least MultiBit and Electrum, probably more. For Android we have BitcoinSpinner and an Electrum based Android client is in development.
The other issue OP mentioned is storage. One of the main strong points of Bitcoin is that it doesn't require trust to 3rd parties for storing money or using money. I know there is demand for services like these but storing bitcoins in an exchange requires people to trust that 3rd party. I have no doubt that currently Mt. Gox is probably one of the safest centralized places to store coins, but it's still centralized. No worries though, there are already solutions to this problem!
Web wallets such as StrongCoin (
https://strongcoin.com/) and the Blockchain.info wallet (
https://blockchain.info/wallet) require very little trust. This is because the private keys to actually use those bitcoins are not stored in their servers. They only have an encrypted version of the key which is useless to anyone if a strong password was used. And when you create a wallet using these services they tell you exactly how strong your password is, so they are fairly user friendly as well. So it's already possible to use Web wallets in a very secure way.
Of course these are still vulnerable to your PC having keylogger trojans etc. and to combat this there are visual keyboards if you are paranoid about security. Even the visual keyboards could technically be hacked so I don't know if anyone has found a convenient way of using bitcoins with absolute 100% security, I doubt there is such a way. But we have come a LONG WAY from the fiasco of MyBitcoin, solutions such as StrongCoin are a major development.
Of course people can trust Mt. Gox and that might work out, but I would not recommend anyone storing all of their money in a place which requires 3rd party trust. Before Bitcoin, there were very few ways to store and use money without 3rd party trust but now there is, so TAKE ADVANTAGE OF IT.