Learn Hardcore Russian: После обеда я возвращаюсь в библиотеку.

Because в with the accusative expresses motion into a place: here it means “return to the library.” If you were already in the library and speaking about location, you’d use в библиотеке (locative).

What does the suffix -ся add to возвращаюсь?

It makes the verb reflexive. Возвращаться means “to return” or “to come back” (on one’s own), not “to return something.”

Why use the present tense я возвращаюсь for a future action instead of the future tense я вернусь?

Вернуться is the perfective infinitive (“to return” with focus on completion), so its future is я вернусь and stresses the act’s endpoint.
Возвращаться is imperfective (“to be returning”) and in present tense can express scheduled or near-future actions or habitual routines. Here it implies a planned or repeated return.

What is the infinitive form of возвращаюсь, and how is it conjugated for я?

Infinitive: возвращаться
Present tense conjugation for я: я возвращаюсь
(Full present paradigm: я возвращаюсь, ты возвращаешься, он/она возвращается, мы возвращаемся, вы возвращаетесь, они возвращаются.)

Can I omit the pronoun я in this sentence?

Yes. Russian is a pro-drop language: the verb ending -юсь already marks the first person singular, so Возвращаюсь в библиотеку после обеда is perfectly natural.

Where is the stress in обеда and возвращаюсь?

Stress falls on the second syllable of both words:
• обе́да (a-BE-da)
• возвраща́юсь (voz-vra-SHCHA-yusʹ)

Why is there no article a or the before библиотеку?

Russian has no definite or indefinite articles. Context determines whether you mean “a library,” “the library,” etc.

Could I say Я возвращаюсь в библиотеку после обеда? Would that change anything?

Yes, you can place я at the beginning: Я возвращаюсь в библиотеку после обеда. Word order in Russian is flexible; moving я simply shifts emphasis slightly onto the subject.