Can you please give some examples of computation of the derived functors $\operatorname{Tor}_1$ and $\operatorname{Ext}^1$ for some simple cases, say $R=\mathbb{Z}$ or $R=\mathbb{Z}[G]$ for some finite group $G$?
Computation of $\operatorname{Tor}_1$ and $\operatorname{Ext}^1$
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homological-algebra
self-learning
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2http://www.math.wichita.edu/~pparker/classes/handout/torext.pdf lists basic properties and some examples. – 2012-02-12
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7All textbooks about homological algebra that I know of include examples. Which have you looked? – 2012-02-12