"Let's go swim together again next month."
またいっしょうに= again, let's go together to (what comes after に)
いっしょうまたに=together (this time), we should again swim
First example implies they already swum together, and また emphasizes it as a repeat (like いつも or よく). The focus is on いっしょう.
Second one implies they haven't swum together, but they have swum separately, so this time they will swim together (again). The focus is on また。
It's almost always いっしょうに(一生に), if any other modifier is used.
I would say no because:
1) Mo means also, not again, so I would say the shade of meaning is different. (It can mean again buuuut...)
2) If you did use mo, it would have to be combined with ni. Mo can only replace wa, ga, and o, otherwise it is used in combination with de and ni.
Okay, I see what you're saying.
In that sentence, "saraishuu" is the topic. It would be "再来週は練習しましょう" (raishuu wa renshuu shimashou) - let's practice next week (with the emphasis of "speaking of next week, let's practice"). In that case, you replace the "wa" with "mo" to change it to "let's practice next week as well."
In this sentence, you can do the same thing if you're treating "raigetsu" as the topic of the sentence (which I was not). "来月は一緒に泳ぎましょう" (raigetsu wa issho ni oyogimashou) - let's swim together next month (with the emphasis being on "speaking of next month, let's swim together"). Then you can replace "wa" with "mo" to mean "let's go swimming next month as well."
Using "mo" vs using "mata" gives me a slightly different feeling about the sentence, but I see how "mo" could work.
来月 (raigetsu) - next month
また (mata) - again
一緒に (issho ni) - together
泳ぎましょう (oyogimashou) - let's (go) swim
The "-mashou" form of the verb is an invitation, usually translated as "let's". You form it by changing the "-masu" ending of the verb to "-mashou".
oyogimasu > oyogimashou