1 is the most concise, 2 is fine, while 3 is a bit off
Let's break things down:
- "I" is the subject of the sentence the thing that is acting on the object.
- "am" is the first-person singular of the verb to be. Importantly, it is an intransitive verb of incomplete predication, which means that it must be followed by a subjective complement that tells us something about the subject.
- "going to" is a modal verb that indicates that the subjective complement will or is planned to affect the object (I) in the future.
- "study" is our first subjective complement - the thing the verb to be is telling us the object is, or because of the "going to" will be. We don't know exactly what sense of study is intended, but it's probably clear from context.
- "and" is a conjunction telling us that there's another subjective complement somewhere up ahead and that I will be both of them.
- "go to the supermarket" I'm not going to break this down in detail, it acts as the second subjective complement in the sentence. The phrase is close to an idiom, meaning I will physically go to the supermarket with the implicit understanding that I will buy groceries and then come home. I should make it clear if I was going for some other purpose. Similarly, if I wanted you to focus on the journey, I should more clearly direct your attention there. I could use the phrase "grocery shopping" and not change the sentence's meaning. OK, it seems I lied and I did break it down.
- "In the evening" is a temporal phrase telling when the being is expected in the "going to" future. It's slightly ambiguous if both of these are happening in the evening; the more natural reading is that you are going to study now, or at least before you go grocery shopping in the evening. If you really mean for us to understand that both of these happen in the evening, you would reverse to the sentence structure and say, "In the evening, ..."
So, with all that out of the way, option 1 provides the most concise information.
Option 2 repeats the subject, the intransitive verb, and the modal verb and might be said for rhetorical emphasis, but if so, it would probably be two sentences. It would be perfectly natural if there was a different subject - "I am going to study and [you are] going to go to the supermarket in the evening."
Option 3 doesn't really work because "to" has made "go" intransitive, and it's sort of flailing around looking for a subject to latch onto. "I" is there, but it's already got the intransitive "am" hooked up, and it is struggling to bear the weight of another intransitive verb. In conversation, it would probably pass by unremarked, possibly even unnoticed - spoken English is pretty tolerant of grammar irregularities. However, in writing, it's just not right.
