Questions & Answers about Я хочу, чтобы ты поехал к обоим полюсам.
Why is it обоим here instead of обоих or обеим?
The word for pole (полюс) is masculine, so we use the masculine/neuter base обо-. The preposition к requires the dative case. The dative masculine form of both is обоим. If the noun were feminine, it would be обеим.
I thought nouns after оба take the genitive singular. Why is полюсам plural?
That genitive singular rule only applies when оба is in the nominative or inanimate accusative case. When оба changes to an oblique case like the dative обоим here, the noun simply matches its case but in the plural. So, both words are dative plural: обоим полюсам.
Why is поехал in the past tense if the trip is in the future?
When you want someone else to do something in Russian, you use хочу, чтобы followed by a past tense verb. Even though the trip hasn't happened yet, the conjunction чтобы always requires the past tense form.
Could I use пошёл instead of поехал?
You would only use пошёл if you expect them to travel by foot. Since going to the poles requires a vehicle or ship, you must use a transport verb like поехал. The prefix по- makes it perfective, meaning you want them to successfully set off and make the trip.