Learn German: Wir besuchen spontan einen anderen Kontinent. - We are spontaneously visiting another continent.

You would place it right after the verb and before the object. For example: Wir besuchen irgendwann spontan einen anderen Kontinent. In German word order, adverbs of time (like irgendwann) generally come before adverbs of manner (like spontan).

Why is it einen anderen Kontinent with those -en endings?

Kontinent is a masculine noun (der Kontinent), and here it is the direct object receiving the action of visiting. This means it must be in the accusative case. The indefinite article changes to einen, and following the rules for adjective endings, the word ander- (other) takes the -en ending to match.

Why doesn't spontan get an ending like anderen does?

In German, words only get grammatical endings when they act as adjectives placed directly in front of a noun (like anderen modifying Kontinent). Here, spontan is acting as an adverb describing the verb besuchen (how we are visiting). Adverbs never take endings.

The English translation says "are visiting", but isn't besuchen just the simple present tense?

Yes! German doesn't have a dedicated continuous tense (like the English "-ing" form). Wir besuchen can mean "we visit", "we are visiting", or even "we will visit" if the context implies the near future. You just use the standard present tense for all of these.