Vervolgens lezen we samen hardop een kort gedicht in de tuin.

vervolgens is a sequence adverb (also called a conjunctive adverb) meaning “then” or “next.” When you place it at the very beginning of a main clause, Dutch follows the V2‐rule (“verb second”), which causes subject–verb inversion. Hence:

• Instead of we lezen, you get lezen we after vervolgens.

Dutch adjective inflection depends on gender and definiteness:

• gedicht is a neuter noun (het gedicht).
• With an indefinite article een, a singular neuter noun’s adjective stays uninflected: een kort gedicht.
• If it were a definite noun (het gedicht), you’d say het korte gedicht.

Both samen (“together”) and hardop (“out loud”) are adverbs of company and manner. In Dutch they typically follow the finite verb but precede the object. You can swap them without breaking the grammar, though you may shift emphasis slightly:

• lezen we samen hardop een kort gedicht…
• lezen we hardop samen een kort gedicht…

Preposition choice in Dutch often matches the mental image of being “inside” an area. A garden is viewed as an enclosed space, so you use in. You would only say op for surfaces you stand on (e.g. op het balkon, op de brug).

Yes, daarna (“after that”) is a common alternative. Nuance:

• vervolgens feels a bit more formal or sequential (e.g. in instructions).
• daarna is slightly more conversational.
  Both trigger the same inversion when placed front-of-clause.

They are near-synonyms:

• hardop is more commonly used in everyday speech to mean “out loud.”
• luidop also means “in a loud voice,” but can sound a bit more emphatic or formal.
  In most contexts you can choose either without confusing your listener.