I am having great difficulty to figure out which word fits best in the following examples.
Do reinigen, sauber machen and säubern fit here as a translation for to clean?
I need to clean my room.
They have cleaned their street from trash.
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In general, all these words have the same meaning and you can use them interchangeably. You'd use any of the words when you clean something. A room, the floor, clothes, a wound, the streets, a house, your fingernails. I consider one of the words slightly more formal, because it's commonly used for any kind of professional cleaning; but it's not formal per se. More about this in a minute.
Here are a few examples:
Sauber machen is pretty informal and used when cleaning, for example, your room or your clothes.
Mach deine Schuhe sauber.
Ich muss noch (mein Zimmer) sauber machen.
Putzen is a common synonym in both cases.
Putz deine Schuhe.
Ich muss noch (mein Zimmer) putzen.
To me, using putzen might connote the use of water or a cleaning agent (Putzmittel). So, when putzen your room, you might do it with water, but when just sauber machen your room, it might just be removing the chaos.
Reinigen could be used in these cases, too. But it's slightly more formal to my mind. Your room service in a hotel will reinigen your room. The laundry service will reinigen your clothes.
Mein Zimmer wurde noch nicht gereinigt.
Ich muss meine Kleidung reinigen lassen.
The noun Reinigung describes the place where you go to let you clothes clean. The noun Straßenreinigung describes the street cleaning.
Still, even for professional cleaning, you can always go with the informal terms.
Mein Zimmer wurde noch nicht geputzt/sauber gemacht.
Ich muss meine Kleidung sauber machen/säubern lassen.
Säubern is more informal, again. I guess I'd use it most likely with clothes or wounds; though, the other words are fine again.
Ich sollte die Wunde säubern/sauber machen/reinigen.
To sum up, you can basically you any of the words when cleaning something. If you clean yourself or your hands, the best word choice, however, is probably waschen. And though, you could use any of the terms above again.
Wasch deine Hände.
Mach dir die Hände sauber.
Putz deine Hände.
Reinige deine Hände.