"I did not study yesterday."
About adding or not the は after 今日, 明日 etc, somebody please tell me if this is correct? I feel like it depends on the question/sentence that led to the one we're translating, for example: If someone asked "what did you do yesterday?" we would add the は because "yesterday" is the topic. If they asked "when did you study?" we would not add the は because "studying" is the topic. So both answers should be accepted because we don't have this context to base our answer on. How far from right is this?
しません is the polite negative version of する meaning "to do"。～ませ～ん alone is not a verb, it's just two auxiliaries that are attached to the i-stem of a given verb to transform it into a polite negative version.
In this case, the i-stem of the irregular する is し and then you attach ～ません to it. しません and しない are both the negative versions of します and する。
Other examples are:
ある → あります → ありません
いる → います → いません
話す → 話します → 話しません
飲む → 飲みます → 飲みません
In this sentence you are also using a past negative form of する, しませんでした is whole. I know is confusing because the past of です is also でした、but it's better if you see しませんでした as one word than a combination of different things. It gets easier later on.
Using the を is also technically correct. べんきょうします (benkyoushimasu) is somewhat special in that it doesn't need an を though. べんきょうをします would basically be like saying, "I do studying." It's not wrong, but it's unnecessary when you could just say "studying" to mean the same thing.
For example, if someone asked what you were doing tonight, you'd normally just answer, "I will be studying," not "I will be doing studying."
Because of the negation. There is no negative past tense form for "shimasu", so the verb is negated normally by adding "-en" and the past tense form of "desu" is added. Therefore, "shimashita" is negated as "shimasen deshita". The English comparison would be that "I studied" would be negated as "I didn't study" rather than "I studiedn't".