Looking at the "Stability Vs ESR" chart you do the following to calculate ESR for a given capacitance.
1) Select a capacitor value in farads.
2) Take the square root of that value to get sqrt(CL).
3) look at where sqrt(CL) value is on the vertical axis of the chart.
4) Go directly right on the chart until you hit the big black diagonal line that marks the region of stability.
5) Wherever you were on that line, go straight down and read the (1/ESR) value off the horizontal axis.
6) Take the reciprocal of that value. That will be the required ESR value.
You don't necessarily need to find a capacitor with the correct ESR. You can always just add a resistor in series with the capacitor to make the recommended ESR.
Note 1 in the datasheet is only recommending a tantalum capacitor on the output. Its not actually saying anything about what type of capacitor to use on the input. The graphs in the datasheet apply only to the ESR of the output capacitor. So you are free to just use a regular 0.1uF X7R type ceramic capacitor. The ESR on the input capacitor can be as low as you like (and you will probably get better performance if it is lower).
Also, the use of Tantalum capacitors on the output is just a recommendation. If you can find ceramic ones that have the right value they will also work. And will probably be much cheaper. You will just need to add the ESR for stability, since the ceramic ones have very low ESR.