For example: tell him to be ready at 6 p.m., I will ( pick him up/take him) on the way home.
Which is the correct option?
Could you explain me and give me one or two examples, please?
There are two possible contexts...
1 - you're taking him all the way back home with you,
2 - you'll be dropping him off somewhere else before you get home.
I'll pick him up on the way home is far more likely. This version works for both the above.
I'll take him on the way home only works for the second meaning. This is because on the way home means during the journey home (you can't take him all the way home until the journey is finished).
It might help to consider these two related constructions...
3: She asked him the way home
4: She asked him on the way home
...where we don't know what she asked him in #2 - all we know is that she asked during the journey.
TL;DR: Both are valid, but #2 has a very specific meaning that usually isn't likely to be intended.
The difference between the two is that take means to transport somebody to somewhere, and pick up means transport somebody from somewhere.
I will take him on the way home.
You might say this if a colleague needs a lift from work, and so you take him to his home, on your way from work to your home.
I will pick him up on the way home.
You might say this if your son needs to get home from school. On the way home from work, you pick him up from the school and take him to your home.