Words such as damit and darum are known variously as Pronominaladverbien or Präpositionaladverbien.
In your examples, they are used as Korrelate, which is a term for words that refer to a clause. For instance, in the following example, daran refers to the clause marked in italics.
Sie hat sich daran gewöhnt, dass das Wetter in Hamburg immer schlecht ist.
Note that in many cases, the Korrelat is optional.
Ich habe mich gefreut, dass du mir geschrieben hast.
Ich habe mich darüber gefreut, dass du mir geschrieben hast.
As a rule of thumb, I would say that for a Korrelat to be optional, the verb in the main clause must be able to stand without the preposition. Compare:
Du hast mir geschrieben. Ich habe mich gefreut.
Das Wetter ist schlecht. *Ich habe mich gewöhnt.
The second example is ungrammatical, therefore daran is not optional with gewöhnen.
Which Pronominaladverb is chosen depends on the context. In the above examples, the verb gewöhnen governs the preposition an and the verb freuen governs the preposition über, therefore daran and darüber are chosen respectively.
The function of Pronominaladverbien is twofold:
The chosen preposition indicates what function the Pronominaladverb (and therefore, the clause it refers to) fulfils in the main clause.
The presence of a Pronominaladverb indicates to the listener to expect a subordinate clause.
With regard to your examples: Es geht um … is similar to gewöhnen in that the preposition, and therefore the Korrelat, are not optional. Therefore, sentences like the following are ungrammatical:
*Es geht nicht nur.
*Es geht nicht nur, neue Wörter zu lernen.
Contrary to Pronominaladverbien, prepositions cannot be followed by clauses.
*Es geht um neue Wörter zu lernen.
With regard to the examples involving beginnen, note that they are constructed differently. In one case, Gespräch is the object and preposition mit is used to indicate what happens at the beginning of the thing indicated by the object, i.e. the conversation.
Er begann das Gespräch mit einer Frage.
Er begann das Gespräch damit, eine Frage zu stellen.
Remember that one function of Pronominaladverbien is to indicate what role a clause has in a sentence; damit is a signal that the clause it refers to is not be interpreted as an object, but "comitatively", i.e. indicating accompaniment.
In the other case, the clause is the object: He started to ask a question, but was interrupted.
Er begann, eine Frage zu stellen, doch er wurde sofort unterbrochen.