Learn Finnish: Nuo ovat harvinaisia kolikoita, joita minä keräilen.

Joita is the partitive plural form of joka. It must be plural because it refers back to the plural word kolikoita (coins). It takes the partitive case because the verb keräillä (to collect) requires a partitive object. A relative pronoun always follows the grammar rules of its own part of the sentence.

Why are the coins in the partitive plural (harvinaisia kolikoita) rather than the nominative plural (harvinaiset kolikot)?

In Finnish 'A is B' sentences, when talking about plural things, you use the partitive plural to talk about an indefinite amount ('Those are some rare coins'). If you used the nominative plural (Nuo ovat harvinaiset kolikot), it would mean 'Those are the rare coins', referring to a specific, complete set.

The verb is keräilen. Is this different from kerään?

Yes, they have a slightly different feeling. Keräillä (which gives us keräilen) means to collect things as a continuous hobby. The base verb kerätä (which gives us kerään) is generally used for gathering things together, like picking berries or collecting trash.

Why does the sentence start with nuo instead of ne?

Nuo is used to physically point out things you can see at a distance ('those over there'). Ne is more often used as the pronoun 'they' to refer back to things you were just talking about, or things that are not physically present to point at.