In German, there are common verbs that you will hear in relation to pain and injury, but also certain colloquial expressions. 

Ist jemand verletzt?

Is anyone injured?

Caption 35, Deutsch für Anfänger: #10 Notfall

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Wer städtisches Recht verletzt hatte, flüchtete zum Domnapf.

Anyone who had violated municipal laws fled to the Domnapf.

Caption 40, Jenny zeigt uns: den Speyerer Dom

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When the focus is really on the pain, the best verb to use is weh tun (the spelling wehtun is also accepted).

Es tut weh. -Der Arzt schaut sich das gleich an.

It hurts. -The doctor will examine it in just a moment.

Caption 13, Nicos Weg: Beim Arzt

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Sie tun mir weh!

You're hurting me!

Caption 2, Die Pfefferkörner: Abi um jeden Preis

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With the verb schaden, the emphasis is really on damage. Below you can also see that the German counterpart of the common English phrase "It can't hurt" utilizes schaden.

Doch ist für Experten wie Schmolz noch völlig offen, ob die Papageien den heimischen Vogelarten schaden.

But indeed, for experts, like Schmolz, it still remains an open question, whether or not the parrots are damaging the native bird species.

Captions 19-20, Freilebende Papageien: Überwintern in Wiesbaden

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Ein wenig neugierig zu sein kann manchmal gar nicht schaden.

Sometimes, indeed, it can't hurt to be a little curious.

Caption 88, Märchen: Sagenhaft Der Zauberlehrling

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Just as die Verletzung means "the injury" and der Schaden means "the damage," der Schmerz is the main German word for pain. Like weh tun, the verb schmerzen is typically used to describe physical pain.

Da Lisas Fuß am Abend immer noch sehr schmerzte, entschlossen wir uns, die Nacht an diesem schönen Platz [entschlossen, Infinitiv: entschließen] unter Bäumen zu verbringen.

Since Lisa's foot still hurt a lot in the evening, we decided to spend the night in that beautiful place [decided, infinitive: to decide] under some trees.

Captions 11-13, Deutsch mit Eylin: Urlaub im Präteritum

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Further Learning
You'll find many examples like this on Yabla German.