Which number is bigger: $\sqrt[10]{2}$ or $1.2$? What is the general method for finding such inequalities? I have some more problems of this kind in the text I am using.

There is no general method. There have been numerous books written on just inequalities. For example, see Inequalities: Theorems, Techniques and Selected Problems. However, oftentimes, simple inequalities like yours may be approached by applying some subtle manipulations, much as mathlove's excellent answer has done. Consider the following example.
Example: Compare $6^{1/4}$ and $4^{1/3}$ without using a calculator. Is one greater than the other?
Answer. Observe that
$$
6^{\frac{1}{4}}=6^{\frac{3}{12}}=(6^3)^{\frac{1}{12}}=216^{\frac{1}{12}}
$$
and
$$
4^{\frac{1}{3}}=4^{\frac{4}{12}}=(4^4)^{\frac{1}{12}}=256^{\frac{1}{12}}.
$$
Since $256$ and $216$ are both being raised to the same power, and $256>216$, we can clearly see that $4^{1/3} > 6^{1/4}$ since
$$
4^{\frac{1}{3}} = 256^{\frac{1}{12}} > 216^{\frac{1}{12}} = 6^{\frac{1}{4}}.
$$

Knowing that it helps to make such manipulations comes with time and practice. Notice that it would otherwise be quite difficult to compare the values in the example as $4^{1/3}\approx 1.5874$ and $6^{1/4}\approx 1.5651$. 
