【1】the first time 主要有以下2个用法

(1)作为名词词组，作句子主语和表语，后面接that定语从句。

This is the first time that she has come here.

这是她第一次来这里

This is the first time that she has experienced disappointment.

这是她第一次感受到失望。

(2)作为连词，引导时间状语从句

Mary was a shy girl the first time I met her.

我第一次玛丽时是一个害羞女孩。

【2】for the first time是个介词短语，意为“第一次，首次，初次”，一般用作状语，修饰谓语动词，可以放在句首，可以放在句末。

For the first time the government's actually put some money up front.  

政府居然预付了一笔钱，这可是有史以来第一次。

For the first time in many years, I got carded.  

这么多年来，头一回有人要我出示身份证。

Several of these rhododendrons will flower this year for the first time.  

这些杜鹃花有些将在今年首度开花。

There is no feeling on earth like winning for the first time.  

初尝胜利滋味的感觉真是无可比拟。

【3】first为序数词或者副词，表示最早的;最先的;第一位的，起初，最早;最先;先，作定语或者状语。

She lost 16 pounds in the first month of her diet.  

她节食的头一个月就减掉了16磅。

When he first came home he wouldn't say anything about what he'd been doing.  

刚回到家时，他只字不提自己一直在做什么。

I do not remember who spoke first, but we all expressed the same opinion.  

我不记得是谁最先发言的，但我们全都表达了相同的看法。

First, tell me what you think of my products. 

先谈谈你觉得我的产品怎么样吧。