You can do two things to upgrade your existing Mac mini:
1) Purchase a retail copy of 10.5 Leopard and install it.
2) Purchase a retail copy of 10.6 Snow Leopard and install it.
You must pay for any "major point" upgrade. Any time the number immediately following the "10." changes (as in 10.4 to 10.5), you must purchase the upgrade. Any "minor point" upgrades (the numbers that follow the major point, like 10.4.8 to 10.4.9) are free and available through Software Update.
Purchasing a new computer will not allow you to upgrade anything on your old Mac mini. The software that ships with a new Mac computer will be incompatible with your old Mac mini, and even if it were compatible, the license agreement prevents you from installing the OS on two Macs from one set of CDs/DVDs. The rule of thumb is one retail copy equals an installation on one Mac. If you have two Macs, purchase two copies.
The Family Pack is different -- it allows you to install on up to 5 Macs that are located in the same household with one DVD.
iLife came separate from 10.5 Leopard, and must be purchased separately, but came with the Leopard Box Set (if I recall correctly, that there was a Leopard Box Set). iLife comes with the Snow Leopard Box Set, but not with the $29 Snow Leopard Upgrade set.