"I took a shower, and then went to the university."
From PuniPuni Japanese:
Asagohan o tabete, gakkō ni ikimashita.
I ate breakfast and went to school.
In the example above, a chronological relationship between the two sentences has been established.
It is obvious by reading the sentence that the speaker first ate breakfast and then went to school.
It probably just hadn't been added to the database yet. Hopefully someone has reported it by now.
The stem form (in this case 浴び) is used to join clauses in literature. It would rarely be used in speech.
The -te form (in this case 浴びて) is used to join clauses in normal speech and is much more commonly used.
The masu-form of the verb is 浴びます (abimasu), but the dictionary form is 浴びる (abiru), not "abu". Like 食べる・食べます (taberu / tabemasu), this is what is called a ru-verb (or "ichidan verb"). It's formed by adding -ru or -masu directly to the stem, 浴び, and to make the te-form you just add -te directly to the stem.
大学 is a university or college https://jisho.org/search/%E5%A4%A7%E5%AD%A6
大学校 is "an educational facility established in affiliation with a government agency" https://jisho.org/search/%E5%A4%A7%E5%AD%A6%E6%A0%A1