one of my classmates told me that this university is very go

两个that 都可以去掉,因为前后两句中that只是单纯的引导词,不充当句子中任何成分.尤其是that引导的宾语从句.
但是句子中this university is very good,和后面的the university is really good 重复感很强.可以将后面的一句用代词省略.
另外那个because有点问题,你的同学说一个学校很好,不是因为他申请了就好,而是因为学校好才去申请.所以句子要把因果逻辑稍微改一下. 另外就是时态问题.
不介意的话,我就帮你把句子稍微改一下
one of my classmates told me that this university which he had applied for was very good, so I searched the information on the internet and found it truely was.

都可以。

第一个可以one of my classmates told me this university is very good,

可以

that都可以去除，因为你的句子不复杂，不用that切分，读者也能分清宾语从句部分。倒是最后一个the university不妨用it替代掉。

两个that在句子中都是作宾语从句的引导词，多可以省略

前面的肯定不能去。因为后面有一句句子。
后面的最好也别去，去的话也可以。