Translation:I borrowed a new book from the library and read it.
When you see a verb in the -te form that comes before the final verb, it (often) signals that you need "and" (unless the final verb is a helping verb, like with "-te imasu" which signals the present progressive tense).
借りて読みました (karite yomimashita)
I borrowed and read it.
買って食べました (katte tabemashita)
I bought and ate it.
図書館から新しい本を借りて読みました (toshokan kara atarashii hon o karite yomimashita)
I borrowed a new book from the library and read it.
You might replace "and" with "so" or "then" or maybe some other slight deviations, but the above translation is the most accurate and faithful to the Japanese.
From Tae Kim:
the te-form alone is used to express a sequence of actions that happen one after another. This will make your conversations smoother as it allows you to connect multiple sentences instead of having many smaller, separate sentences that are often too short.
朝、起きた。そして、朝ご飯を食べた。そして、学校に行った。 Morning, (I) woke up. Then (I) ate breakfast. Then, (I) went to school.
朝、起きて、朝ご飯を食べて、学校に行った。 Morning, (I) woke up, ate breakfast, and went to school.
Puni Puni gives a simple and clear explanation that is worth a read.