How do you use two verbs in one sentence in Japanese?
I know how to write a basic sentence like: I go to school, but I don't know how to write sentences like: I go there to see her. How would I write that sentence?

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Answer

I think there are many ways to use two verbs in a sentence.
Anyway, your example would be translated into like this:
#1:彼女に会うために学校に行きます。
#2:彼女に会いに学校に行きます。

The case of writing a purpose to do something:
in order to do
=>
#1 verb(dictionary form) + ために（or　ため）
or
#2 verb（stem of マスform)　＋　に

#1 食べ物を買うためにスーパーに行きます。
#2 食べ物を買いにスーパーに行きます。

These have the same meaning.
However, as Jaida-san wrote, #2 phrase can only be used with movement verbs like 行く、来る、戻る、帰る.
#1 phrase can always be used.
So may be it's easier for you to start with #1 phrase.

And if you learn a Verb stem of マスform, you can make another way to use two verbs like Garrett-san wrote.
I finished reading a book.
本を読み終わりました。
I started learning Japanese.
日本語を勉強し始めました。
These are:
verb (stem of マスform)　＋　verb（meaning to start, finish)
読み＋終わる＝読み終わる
勉強し＋始める＝勉強し始める

I hope this helps you!

Answer

From the information of you all, we can use even three verbs in one sentence now.
Like; 私は彼女と話すために英語を習い始めた。I started learning English to talk with her.

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Another way is to change the ending to an "i" ending (み、び, etc) and then add the next verb. I don't know if this works all all types of verbs though. 
読む> 読み終わった
シャワー浴びる> シャワー浴び終わった

Maybe you could be more specific in how you want to combine verbs and provide a sample sentence in English.

Answer

Going in order to do something you can use Verb stem+にきます However this form only works for 行きます、来ます、and 帰ります
彼女をあそこに見に行きます "Im going there in order to see her"

Im going to the library to study 図書館に勉強にいきます。

Im not native so there might be a better way to say it, but thats one way i know of