Translation:I go to the hospital.
you are absolutely correct. the verb "ikimasu" is "i go" or "i will go". the word "going" is a progressive verb so it should be itteimasu "行っています" "びょういんに行っています" "itteimasu" and not "ikiteimasu" as iku 行く is an intransitive verb. iku being the ru form of ikimasu. duolingo is good for teaching words and sentence structure, however I suggest you find an alternative source to learn grammar.
Be careful. Present continuous for verbs of state change (行く, oddly, included) means something completely different in Japanese than in English.
When used with action verbs, such as 食べる, it means just a continuous action - 食べている, "I am eating".
But when used for verbs like 行く or 来る, it means change of state, completion of the movement. Therefore 田中さんが来ています doesn't mean that Mr. Tanaka is on his way, it means that he is already here. Similarly, 行っています doesn't necessarily mean that the person is currently physically moving, it means that the person has left for his/her destination.
If you want to emphasize the continuation of the action you have to use a special construction for these verbs, or just use ～途中です ("I'm on my way").
Agreed on the alternative source part. Duolingo's Japanese course is still very much work in progress.
I asked about this matter and got an answer from a contributor: https://www.duolingo.com/comment/27657664$comment_id=27665614
I'm Irish, I'd say "I go to hospital" to mean "I attend a doctor/service at a hospital" and "I go to the hospital" to mean I'm physically travelling to a hospital. Like "I go to hospital for my antenatal appointments" but "I go to the hospital on Saturdays to visit my sick mother".
I'm going to school means "I am a student of the school." I'm going to the school means "I'm traveling to the building in which school is held." This can be for any purpose: PTA meeting, to see a play, w/e. This is the same for hospital outside the US. "I'm going to hospital," as a patient. "I'm going to the hospital," for any reason that requires traveling to that building/campus. "I go to ___" works the same way. This is also used for church and probably other situations in which a building & activity use the same word.
They are not substitutes, Duolingo is just bad at explaining them. Present progressive/continuous in Japanese has different meanings depending on context and type of verbs, check piguy3's response or mine. It's not the same thing as in English and it's one of the pitfalls of teaching Japanese - I understand the course creators consider this kind of sentences important (and they are right) but at the same time they are confusing beginners because Japanese present continuous is not explained properly.
I realize "I'm going to the hospital" is a different tense, but as a native English speaker "I go to the hospital" sounds unnatural to me. It sounds as if I'm saying it's part of my routine or a regular occurrence, when in fact for most people going to a hospital would be an out of the ordinary occurrence.
I think it is either a future state or a repeated action state, meaning "I will go to the hospital" or "I'm going to the hospital" - in the sense that the speaker is regularly going there. Without more context, we can't tell.
However what we can tell is that at the moment of utterance, the speaker is not currently at the hospital or on route to the hospital.
If he was at the hospital right now, then he would say something like 今病院にいます ("I'm at the hospital right now") or 病院に来ました ("I arrived at the hospital"). So the speaker is talking to someone outside the hospital about him going to the hospital and he is also probably not about to go there right now - because then the most natural way of saying that would be 病院に行ってきます ("I'm going to the hospital and back").
He is not on route either because then he would probably say 病院に行く途中です ("I'm on my way to the hospital").