Practically, few things in real life have convenient closed-form calculations.
Instead, you price some exotic, then you bump the various inputs, one or several at a time, up and down, by various small amounts, and re-price. There are seldom any short-cuts. (Autodiff can sometimes be a shortcut.)
This Wikipedia article actually has a good list of commonly used risk mesures: https://en.wikipedia.org/wiki/Greeks_(finance)
During model validation and ongoing model performance monitoring, you figure out which risk measures are important (or can become important under plausible large market moves). Then you put limits on them and calculate them a lot. There's no glamorous math here, just lots of brute force automated calculations.
Edit: thanks KermittFrog for reminding that different risk measures can be used for different purposes. Here's an example which actually involves some math. Suppose you hedge your interest rate risk with ED futures until 10 years and IR swaps after 10 years. You fit your IR curve from the hedging instruments. You bump each instrument and refit the IR curve. You reprice each instrument in your portfolio under each bumped IR curve. The resulting sensitivities tell you what notionals of hedging instruments you need to add to the portfolio in order to flatten the IR risk. But suppose further that you want to see the sensivities to IR swap rates from 1 to 10 years, for market risk limits monitoring. Since you don't use these swap rates to fit your IR curve, you can't just perturb them. But you can calculate how these swap rates change then the ED futures change, and multiply the ED futures sentivities of your portfolio by an inverse Jacobian to get a good estimate of the sensitivities to 1-10 swap rates.
As for the book question, I should mention Prof. Carol Alexandar's 4-volume Market Risk Analysis, which is probably an overkill. There is also a discussion of Greeks of exotic options in Chapters 7-9 of Leonardo Marroni, Irene Perdomo. Pricing and Hedging Financial Derivatives: A Guide for Practitioners.