Should this verb be in the third-person singular form, the infinitive form, or the present participle form? [duplicate]

Your sentence has a main clause and two subordinate clauses, and so three verbs total.

The main verb of your sentence, by which I mean the matrix clause with a subject and a tensed or finite verb, is was, which is the singular past tense of the copular verb be. But neither subordinate clause here can be a finite verb like explores or watches; either of those would be ungrammatical. Only non-finite verb forms are allowed for your situation because those clauses are core arguments to other verbs.

Whenever you have clauses serving in the direct object role of another verb, as here you have with the verb watching, these must be non-finite clauses, not finite clauses with tensed verbs. In other words, they must not be inflected for person, number, or tense.

The two possible non-finite clause types able to serve as the core argument of another verb are those that use the -ing form of the verb and those that use the root form of the verb. You sentence happens to have both kinds.

• The subject of the main verb was is a non-finite clause of the first sort:
  Watching my friend explore the environment
• The object of the that clause’s verb watching is a non-finite clause of the second sort:
  my friend explore the environment.

This is sometimes called a bare infinitive because it doesn’t require the normal to particle that infinitives usually require, let alone the for-to complementizer needed when the infinitive clause has an expressed subject, especially when that clause is the subject of a tensed verb.

The lack of specific marker for the infinitive happens with sense verbs, so verbs like watch, see, hear. Two different possibilities exist when choosing a clause as the complement of one of these verbs. These two are not interchangeable because they mean different things. An infinitival clause lacks the continuous aspect that a gerund clause provides.

1. Watching my friend explore the environment with only a sense of wonder was my favourite part of the game.
2. Watching my friend exploring the environment with only a sense of wonder was my favourite part of the game.

A shorter sentence may better illustrate the distinction in aspect.

3. Last night I heard my brother come in through the upstairs window.

4. Last night I heard my brother coming in through the upstairs window.

Do you see the difference there? The second emphasizes the progressive aspect of an ongoing action. You could easily add when the phone rang to the second sentence — but less felicitously (if at all) to the first sentence. It would not make sense.