Bootmii is not a backup loader; it is a pre-boot program that provides a level of security when hacking because it can be used to reverse changes or make changes to the Wii even when the Wii is bricked. That being said, everyone is advised to install Bootmii as boot2 if possible.
Softmodding to run backups is fairly simple. Once you have the Homebrew Channel installed, install a cIOS (Waninkoko's cIOS rev17 and Hermes' cIOS v4 are both good choices) and a WAD Manager, then obtain the app or wad for a backup loader. You can use a disc loader (NeoGamma, Softchip) or a USB loader (Configurable, USB Loader GX, WiiFlow).
Backups run from any of these loaders generally have the full features of the original game, same as if you used your modchip. They can play online, use the same game savefiles, etc. although some games may have compatibility issues with peripherals or SD cards.