"Сходить" means that you go somewhere and then return back. Compare: "Я ходил на кухню" means that you went to the kitchen and doesn't tell us what did you do after; "Я сходил на кухню" means that you went to the kitchen and then returned. In your example about the theatre there is no much difference what word to choose - "Сходим" or "Пойдём". You could say "Давай пойдём в субботу". But you should know that in some expressions better to use word "пойти", in others "сходить". It depends on tradition. We would say "Давай пойдём в субботу", but "Может быть, сходим в субботу".

Возможно вы имели ввиду "пойдём". Да так тоже можно сказать "Может пойдём в субботу?"

You can use "поедем" or IF: - you live in another city. - you want to empasize means of transportation. E.g. You: "Давай поедем на машине?" Таня: "Давай. Тогда поедем завтра, сегодня машина в ремонте". As for why.... With в театр, цирк, кино, зоопарк, .... в гости, etc. we traditionally prefer word 'пойти/сходить'. I don't know why:) May be, when we say this we think NOT about way to there, but about what we are doing there. So when we say "идти в театр" we imagine "chosing a proper dress, buying tickets, taking seats, watching the play" - and this becomes the (new!) 'meaning' of the word ходить.

"сходить" is one of the way to say "go", "пойдем" and ect