To get this out of the way first: It should be sollten (Konjunktiv II). And wirksam does not really work. I am not quite sure what effectively is supposed to mean here, so I am going to stick to effektiv as the other answer does.
Now to unterrichten. This is a slightly strange verb, as it can take an accusative object to either indicate who is taught
Ich unterrichte Schulkinder.
or what is taught
Ich unterrichte Mathematik.
You cannot have both as an accusative object, though. (Now that I think of it, this is similar in English.) If you want both then it is jemanden in etwas (dative) unterrichten.
Ich unterrichte Schulkinder in Mathematik.
Therefore, your first sentence would have to be
Wir sollten Schulkinder darin unterrichten, Handys effektiv zu benutzen.
Note that this sounds stilted, and that it describes more an ongoing process of teaching and not so much that something has been learned finally afterwards.
Let me mention the alternative of using a noun:
Wir sollten Schulkinder im effektiven Gebrauch von Handys unterrichten.
So let us look at other verbs. There is lehren. Now this is a really strange one, because it can take two accusative objects, jemanden etwas (accusative) lehren. Therefore, you can now use the construction that you wanted
Wir sollten Schulkinder lehren, Handys effektiv zu benutzen.
I do not really recommend this, because this use of lehren is somewhat dated. So let us move on to jemandem etwas (accusative) beibringen. This has an accusative and a dative object, so it is less exceptional.
Wir sollten Schulkindern beibringen, Handys effektiv zu benutzen.
This sounds most natural.
To your other variants:
Wir sollten Kinder unterrichten, um Handys effektiv zu benutzen.
This is grammatically correct, but it seems to say that teaching children will enable us to use phones better. Not what you want.
And as to word order in the correct sentences:
Wir sollten Kinder Handys effektiv zu benutzen beibringen.
I cannot think of a reason why this would be incorrect, but it does not read well, do not use it. Again, using nouns is an option.
Wir sollten Kinder den effektiven Umgang mit Handys beibringen.