Wouldn't this save me some fees when I spend the coins?

Why shouldn't I use the same address for every transaction?

To be more precise, the reuse of bitcoin addresses exposes both the initial user holding said funds and everyone else who becomes a future recipient of that bitcoin balance. Anyone who is actively monitoring the bitcoin blockchain to link identities to transactions will pounce on such opportunities with relative ease. Every time one uses the same address, any transaction linked to it in the past, present, and future will have their history exposed to prying eyes.