How do you politely ask for things in Mandarin?
Politeness in Mandarin is expressed by using different politeness elements, which include politeness characters, wording or manner of speaking. Usualy one element is enough in one sentence, because using 2+ elements in the same sentence may sound too polite, causing a feeling of distance.

Would you please pour me some water? 请给我倒些水好吗？

Note: it is different from 我想喝水. Because 我想喝水 is not a request. If you tell me you want to drink some water, I will think, ok, no problem, go ahead and get some water for yourself, as you didn't ask me to get some water for you.
When making a request in Chinese, usualy we don't say 谢谢. Instead, we say 谢谢 when the request is either accepted or fulfilled.

A:您能帮我给他带个口信嘛？ B:可以。A:谢谢。---Request accepted but not fulfilled yet.
A:请给我倒杯水好嘛？ B:给你。A:多谢。---Request fulfilled。

Here are some examples of making polity requests in Mandarin:

您能帮我个忙嘛？ ---Politeness character 您 is an equivalent personal pronoun of 你 in Chinese showing your repect for others, including the elder, your superiors, and so on.
请帮我个忙。---Politeness character 请 is used here to show politeness when asking for a favor
我看，也只有你能帮我这个忙了。 ---Wording, which makes a request indirectly.
帮帮忙吧。 ---Manner of speaking, in a supplicatory way.
可怜可怜我们母子俩吧。---Wording + manner of speaking， which instigates pity.

By the way, the following are the expressions to use when someone says 谢谢/多谢 to you:

别客气 / 甭客气 / 不用客气 / 客气啥 / 太客气了 / 没事儿 / 见外啦 and a lot more.