"Also, today is rainy."
また means "also" in the sense of "again"; therefore 今日はまた雨です kind of implies that it has been raining quite recently too (e.g. yesterday or other days this week).
も is a plain "also/and", and lacks that temporal connection. You could be talking about some historic downpour and then go 今日も雨です: "today it also rains".
As far as i can tell, your sentence could mean "Today it's still raining. " I do feel that the sentence in the exercise isn't translated well. I hope we can get clarification on this exercise soon.
また can mean "also/and", as well as "again/another time". The two are mostly distinguishable by where it's placed in the sentence. In the middle, it often means "again" or "another time". E.g 今日はまた雨です "Today it rains again". At the beginning (like we have here) or after a comma, it is more like "Plus/moreover, ...". E.g. この料理（りょうり）はおいしく、またやすいです。"This dish is delicious, as well as cheap".
そして is more like "also/and" in the sense of "and then/after that, (X happened)" with one thing following another.
そして more appropriately means "and then," where it infers that the previous action occurs before the one after. また when used as a conjunction like this case, it means "in addition" or "also," where it is used to suggest an idea or fact that is usually less important than the previous one.
I left です off and got a wrong answer. That seems to be acceptable in some answers, but not in others. です/でした are pretty much a formality unless the tense is in question, right? Or, I guess it's also necessary if a question is being asked ( because of ですか). But in an example like this, shouldn't it be optional?
This is not true. A sentence can end on a noun. For example, 元気 is a valid sentence. The state of being is implied as well the subject. Using the declarative だ just emphasizes the state of being, it is not required. In some situations it can even be seen as rude. です is alsk not required, however it is polite to do so, a common courtesy to people you do not know, authority, etc. so its best to get in the practice of using it now.
I think it has a different meaning. そして means "then." また means "in addition" or "also." Compare these
- Then, it is rainy. (It is like telling something happens first, then something happens next.)
- In addition, it is rainy. (It is like something happens, and some other things which is related in some way happens, not necessarily in chronological order.)