Translation:I like going to foreign countries.
This is a very good question and there is no easy answer unfortunately. In some cases, both can be used. In other cases, only one can be used. Here are some tips found: ・「の」専用 ( Strictly using 「の」)：「～のを見る／聞く」e.g. Anneが歩いているのを見た。 ・「こと」より「の」を取りやすい(More likely to use「の」than 「こと」) ：「上手だ」「下手だ」「好きだ」「嫌いだ」「いやだ」e.g. Alexは料理をするのが好きだ。 ・「こと」専用( Strictly using 「こと」):the sentence form of「XはYです」e.g. Yokoの趣味は、本を読むことです。
海外 かいがい means overseas. I guess they are used contextwise or own preference. Generally 外国 is learned first. I think because this counts for everyone everywhere (widest use) and 海外 has already a more limited use, because of it's meaning。I guess Japanese use them interchangeably. This is just a bit of guesswork sucked out of my thumb though.
Given the English sentence, は would be OK, but the meaning and situation where you'd use it is slightly different from が. Think of the sentence as the answer to a question: someone could ask "What do you like to do?" or "What do you think about going abroad?"
In English, both questions can be answered with "I like going abroad", but in Japanese you'd answer the first with 「外国に行くのが好きです」 ("Going abroad is what I like to do"; the focus is on going abroad) and the second more likely with 「（外国に行くのは）好きです」 ("Going abroad? I like it"; the focus is on the opinion; note that the first half of that can be left out in both English and Japanese since it's already clear you're talking about going abroad)
好き is equivalent to "like" or "be fond of". In English, when you say "I enjoy traveling" you also mean "I like traveling". But there's a slight different nuance there.
The same happens in Japanese. So Duolingo is trying to stick to the more accurate English equivalent, but you may try reporting it so they include that option.
I think, however, "like" should be the preferred translation here.