Die Mitgliedschaft in der Bibliothek ist kostenlos, aber eine Anmeldung ist notwendig.

The preposition in can govern either the dative or the accusative case.

• Use the dative (here der Bibliothek) when you talk about location (where something is).
• Use the accusative (i.e. in die Bibliothek) when you talk about movement toward a place (where something is going).
  Since you’re stating where membership exists (location), you need the dative: in der Bibliothek.

Because kostenlos is used predicatively after the verb sein (“ist kostenlos”), not attributively before a noun. Predicative adjectives in German always appear in their base (undecorated) form:

• Predicative: Die Mitgliedschaft ist kostenlos.
• Attributive (before a noun) you would decline: eine kostenlose Mitgliedschaft.

Here Anmeldung is a countable noun meaning “a registration” or “one registration.” You need an indefinite article (eine) to indicate one instance of that noun. Omitting the article (or using the plural Anmeldungen) would change the nuance:

• eine Anmeldung = one registration is necessary.
• Anmeldungen (plural) = registrations in general, which doesn’t fit this context.

aber is a coordinating conjunction, so it does not send the verb to the end (unlike subordinating conjunctions such as weil). It simply connects two main clauses, and each clause keeps the normal German word order (verb-second). Thus:

1. first clause: Die Mitgliedschaft (subject) – ist (verb) – kostenlos.
2. second clause: aber (conjunction) – eine Anmeldung (subject) – ist (verb) – notwendig.

Yes, Registrierung (from the verb registrieren) is also correct and understood. However:

• Anmeldung is more common in everyday German, especially for library or event sign-ups.
• Registrierung sounds a bit more formal or technical, often used in IT contexts (e.g. website registration).