If I want to use two mods together, e.g. SinglePlayerCommands and TooManyItems (which don't state compatibility with one another), how can I tell if they will work together (and not break Minecraft - forcing me to re-download a fresh copy)?
There is no hard-and-fast rule other than simply installing them both and seeing if it blows up. However, there are some tricks for guessing ahead of time.
Firstly, mods which use the same API/framework/loader (e.g. ModLoader or Minecraft Forge) are much more likely to be compatible with one another than non-framework mods. This is because those APIs are created specifically to help prevent mods from conflicting with one another. (However, it's still possible — even easy — to create conflicts.)
Mods which use patchers are also somewhat more likely to be compatible with other mods if installed last. (Patchers work by replacing small bits of Minecraft's code rather than entire files.) However, patchers are few and far between.
The best rule-of-thumb indicator of whether two mods will conflict is to compare the list of
.class files they contain. If there are no files which exist in both mods, chances are good that they'll work just fine together. If there are one or more files which exist in both, however, there will very likely be a conflict.
Finally, you can check the mods' documentation to see if they have notes about compatibility. Sometimes, there will be special instructions for getting two mods to work together which otherwise conflict or descriptions of what functionality the conflict will break. (e.g. TooManyItems and Convenient Inventory; ControlPack and OptiMine)