Learn Hardcore Japanese: tosyokan de mezurasii hon wo sagasimasu.

に is used for destinations or locations of existence:
• 図書館に行きます (I go to the library) – destination
• 本が図書館にあります (A book is at the library) – location of existence

What role does を have in 本を探します?

を is the direct-object marker. It tells you that 本 (“book”) is the thing being searched for by the verb 探します.

Why does 珍しい come before 本 with no particle in between?

Japanese adjectives (both i-adjectives and na-adjectives) directly modify nouns without any extra particle. Here, 珍しい (an i-adjective) simply precedes 本 to form “rare book.” English also places adjectives before nouns, but Japanese grammar doesn’t require any connector like “no.”

What type of adjective is 珍しい, what is its reading, and how does it change in different forms?

Reading: めずらしい (mezurashii)
Type: i-adjective (ends in い)
Conjugations:
• Present (attributive): 珍しい本 (“rare book”)
• Past (attributive): 珍しかった本 (“book that was rare”)
• Negative (attributive): 珍しくない本 (“book that is not rare”)
• Adverbial: 珍しく探します (“search … rarely,” though this is rarely used with 探します)

Why is 探します in its -masu form, and what’s its dictionary form?

探します is the polite present/future form used in everyday conversation.
Its dictionary (plain) form is 探す.
Other related forms include:
• Plain: 探す
• Te-form: 探して
• Polite negative: 探しません

Why is there no subject shown in this sentence?

Japanese often omits the subject when it’s clear from context. Here, the speaker (“I” or “we”) is implied, so you don’t need to say 私は図書館で… unless you want to emphasize who is doing the searching.

Can I change the word order to 珍しい本を図書館で探します?

Yes. Japanese word order is relatively flexible as long as the particles mark each role correctly. Both orders mean the same thing:

図書館で珍しい本を探します
珍しい本を図書館で探します
Putting 図書館で first is more common, but your version is also perfectly grammatical.