"Let's do the laundry and then clean up."
Think して as "and" and それから as "from that point on" or "then", so you have "lets do the laundry and then clean up". The て means the previous verb is conjugated like the last, otherwise you would say: Sentakushimashou, sore kara soujishimashou. That sounds unnatural, dont you think?
The person I asked (N1) said, "It's a short casual version of it, yes. Kind of like how you can casually say でしょ instead of でしょう. I would avoid using it because it isn't as polite." (I'm not sure if I'd call it slang, but you're right in that they aren't the same in terms of politeness)
I searched for a while and didn't find much, so I asked some more advanced learners, but they haven't replied yet. For now, you'll have to make do with the dictionary entry I found: ましょう (expression) 1. I'll (Polite, see also ます, used to express volition; ましょ is colloquial; まひょ is ksb). According to my understanding of the line, ましょ is acceptable in speech and not used in extremely polite situations. I'm unsure if it applies to the other definitions of the word (let's, etc.), but it likely does. (https://jisho.org/search/%E3%81%BE%E3%81%97%E3%82%87)
Thank you so much. I still have some doubts, but your comment is very ¿helpful?¿helpfull? ¿Helpfully?; in my mind, ーしまちょう is like «let's do ( right now ), and して is like do ( order more than other thing); then the sentence here is a little weird to me and my own structure lenguage because i think in it like this : せんたくしましょう、それからすおじして。 or すおじして、それからせんたくしましょう。but the good sentence is せんたくして、それからすおじしましょう。can you see what i am trying to ¿transmit?( transmit is a word, i do not know). If you can understand me, please help me, because i think my sentences here are not totally wrong, but maybe yes.
I can understand you, don't worry :)
洗濯しましょう。それから、掃除して。 (Sentaku shimashou. Sore kara souji shite.)
Let's do the laundry. Then you clean. (We will both do the laundry, but only you will clean.)
掃除して、それから洗濯しましょう。 (Souji shite, sore kara sentaku shimashou).
Let's clean and then (let's) do the laundry. (We will both clean, and we will both do they laundry. We will clean first. When we finish cleaning, we will do the laundry.)
洗濯して、それから掃除しましょう。 (Sentaku shite, sore kara souji shimashou.)
Let's do the laundry and then (let's) clean. (We will both do the laundry and we will both clean. We will do the laundry first. When we finish the laundry, we will clean.)
The -te form of the verb (して) can have different meanings, depending on where it is placed in the sentence.
掃除して！ (Souji shite!)
Clean! (it's an order)
掃除して、洗濯しましょう。(Souji shite, sentaku shimashou.) = 掃除しましょう。洗濯しましょう。(Souji shimashou. Sentaku shimashou.)
Let's clean and then (let's) do the laundry.
When the -te form verb comes in the middle of the sentence, it is not an order. It is connecting the sentences together. The -te form verb happens first. The other verb happens next.
掃除して、洗濯しました。 (Souji shite, sentaku shimashita.) = 掃除しました。洗濯しました。 (Souji shimashita. Sentaku shimashita.)
I cleaned, and then I did the laundry.
食べて、行きましょう。 (Tabete, ikimashou.) = 食べましょう。行きましょう。 (Tabemashou. Ikimashou.)
Let's eat and then (let's) go.
Ou yeahhhh , i am learnig!!!!!! So, i am not totally wrong, just the meaning, both or just me/you doing one or another. Shite,shimashou-shite,shimashita- shimashou,shimashou- shimashou,shite-shite,shite. Now i can see why shite first and shimashou finish. Muy agradecido. Thank you very much, 先生~さま sama is like god,isnt it? The super respect from me to you.
Japanese Honorifics -dono: the greatest respect given. I've never seen it used accept when the manga is set in the past. -sama: just below that, also used mainly in the past, but jokingly in the present time when a character is being bossy and giving orders impolitely. -san: Used with almost anyone. Just met someone? Use -san. -kun: used casually amongst school boys and older best friends. -chan: Used casually amongst school girls. Older people use it when talking to their lover and to other peoples children. -sensei: used with teachers, professors and doctors. Senpai- "upper class" Kohai- "lower class" both of these refer to workplaces and schools, not social standings. If one person addresses another as senpai then it's probably someone in her 2nd year of school talking to another in her 4th year. No honorific-family, or someone so close that you consider him as such. People normally use family names when addressing someone. Given names are used when people are closer.
This is a good explainer: https://www.punipunijapan.com/making-proposals-mashou/
If you look up 洗濯, say, on Jisho.org, you'll find that it's listed as both a noun and a suru (する) verb. This means that both 洗濯する (this is how suru verbs are formed) and 洗濯をする (this is how we say "do (noun)") are valid forms of the verb. Additionally, を is often omitted in speech, so what little difference there was is irrelevant. Edit: I didn't mention your second example, but シャツを洗濯して uses 洗濯 as a suru verb. When using the noun form, の is required (it connects two nouns): シャツの洗濯をして. (I'm not completely sure about that, though; if you want a more reliable source I will find one)