Should I use alternate or economy picking?
Alternate picking by default for riffs and scale runs. Economy picking or sweeps when a line travels across strings in one direction, like an arpeggio. Learn both and choose per phrase.
The difference
Alternate picking strictly switches direction on every note: down, up, down, up. Economy picking keeps going in the same direction when the next note is on the next string, so two notes on neighboring strings can share one motion.
Why you need both
If you only economy-pick, your string changes never get strong, and metal riffs are full of them. If you only alternate pick through a six-string arpeggio, you're fighting the geometry. Most metal rhythm is alternate or downpicked. Economy and sweeps show up mostly in leads.
Start with alternate picking and the string-crossing drill. Add sweep picking once your alternate picking is even.