I'm trying to wrap my head around Greeks, and I'm getting a little bit confused. For example, let's say my portfolio holds a long 5 month ATM call with strike \$20, and short 2 month OTM call with strike \$60. Now, if my underlying rises to \$40, what happens to my $\Delta$, $\Gamma$ and $\nu$ega exposures? I'm not exactly sure how much information I can say looking at the Black-Scholes formula since I don't have information about $\sigma$, so what can I say about the response of the Greeks to this underlying change?
The greeks are non-linear and only give you the instantaneous rate of change. The larger the change in underlying is, the less accurate the change based on the greeks will be. A doubling of the underlying will certainly not be predictable by the greeks alone. Delta and Gamma can be used to estimate the new price using a second-order taylor series approximation:
$P \approx P_0 + \Delta * dS + \frac12 * \Gamma * (dS)^2 $
But it's still an approximation, and with large changes in S it may be significantly different than the actual change.
But you asked about the effects on $\Delta$, $\Gamma$, and vega. Gamma will give you an approximate change in delta ($d\Delta \approx \Gamma*dS$). There are greeks that will tell you the change in Gamma and Vega when the underlying changes (Speed and Vanna, respectively), but they are not as widely used.
Of course, the practical way to calculate the difference based on a change in the underlying is to reprice the option with the new spot price, but that assumes that the volatility is the same, which is probably not true in reality (large changes in the underlying often coincide with large changes in implied volatility)