Can I save mining fees when doing transitive transactions?
Probably not. Clients can set whatever rule they like for transaction fees, but they normally adhere to the Satoshi client rules described at https://en.bitcoin.it/wiki/Transaction_fees. However, note the section on transaction priority; if you have an address holding some old coins, it's quite likely a transaction out of that address will have high enough priority for you to avoid transaction fees by default.

To implement what you describe, you would need a client that does something like hold all low-fee transactions in a small pool and watches for new transactions out of their output addresses. That's certainly possible to implement, but it seems to me that it would be the sort of service you'd pay a premium for, not receive a discount on.