The only way to speed up the first confirmation is for you to (not guaranteed, but generally works) pay a higher transaction fee or for you to own a significant amount of the hashing/mining power and implement rules so that your mining farm always includes your transactions. Keep in mind that even if you owned 100% of the hashing power (of which you always prioritised your own transactions), the best you could ever get is your transaction in the next block which will take 0 to 10 minutes.
Note that once your transaction is included in a block, it's generally 10 minutes per confirmation after that (unless the block chain forks and your transaction needs to be re-included in the new chain), so it would take from 50-60 minutes for 6 confirmations anyway.
At the moment, If I pay a transaction fee of 0.0003BTC (30 cents!) my transaction is normally included in the next block. A fee of 0.0001 can take hours.