Can someone tell me how to put two verbs together to create a new meaning?
For example, how do you make verbs like 求めて叫ぶ (to cry for) and such? How do you put them in a sentence to covey the message that, in this case, they are shouting while requesting something?

Question

Answer

In this case the meaning would be "〜するために".

And for this verb, you would need object; 〜を求めて 〜する。

Example: 助けを求めて叫ぶ。
Meaning: 助けを求めるために叫ぶ。(They scream to ask for help.)
Example: 自由を求めて戦う。
Meaning: 自由を求めるために戦う。（They fight for the freedom.)

And another suitable verb is 探す, for instance.

Example: めがねを探して歩き回る。
Meaning : めがねを探すために歩き回る。（They walk around to look for glasses.)

This rule does not apply for all verbs. 
You can use て form in many ways according to the context. 
I recommend you to learn with some example sentences. :)

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