There are a few ways you can do this.
The first method is knowing/learning/memorizing periodic trends. Looking only at the periodic table, we can see that there are certain trends in specific columns. I have labeled these trends for you in the attached image. So, from these trends we can say that phosphorus will have 5 Valence electrons (Ve-) because it is in the 5A column. Valence electrons are the electrons on the atom's outermost electron shell.
The other way you can figure this out is by knowing how electron shells work. In the first electron shell 2 electrons are present. Each shell after that fills up until it runs out of electrons or has 8 electrons. Once a shell has 8 electrons, the next shell starts to fill.
Knowing this, we can draw a picture of the shells using the atom's atomic #. Which in this case, is 15. This number tells us how many protons that element has. The number of protons is often equal to the number of electrons. So, we can assume that Phosphorous has 15 electrons total. If you drew a picture, there would be 2 e- in the first shell, 8 in the second shell, and 5 in the outer most shell. (now we have used up all the electrons.)
Instead of drawing the electron shells, if you know the theory you can use this shortcut most of the time:
(atomic#/ total#of e-) - (the # of e- in the first shell) - (the number of electrons in full shell) = a number less than or equal to 8 (if your answer is more than 8, fill up another shell by subtracting 8 electrons again)
For this example:
15 electrons- 2 electrons- 8 electrons= 5 Valence electrons
