Do vega and gamma always have the same sign (ie both positive or both negative)? Under what circumstances can they have opposite signs?
Usually vega and gamma go in the same direction, but you can have opposite exposure in a calendar spread.
For an ATM option, vega decreases closer to maturity while gamma increases. If you implement the following:
-long a 1 month ATM option
-short a 2 months ATM option
you should be long gamma and short vega.