We see the word che meaning "that" or "which" all the time in sentences. It's a very common conjunction. 

Ad Ercolano, c'è un pomodoro che è diventato simbolo di un'importante voglia di cambiamento.

In Ercolano, there is a tomato that has become a symbol of an important desire for change.

Captions 21-22, Pomodori Vulcanici Pomodori del Vesuvio - Part 7

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But che does more. Here is a lesson about using che to say things with simplicity, a great asset when you're just learning. It helps make conversation. Here, it means "how."

Che carino, Però adesso devo scappare, altrimenti mio fratello mi uccide.

How sweet. But now I have to run, otherwise my brother will kill me.

Caption 29, Il Commissario Manara S2EP9 - L'amica ritrovata - Part 11

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Che can also mean "what." See this lesson. 

Scusa, ma io che ci faccio qui? Non conto niente.

Sorry, but what am I doing here? I don't count for anything.

Caption 3, Moscati, l'amore che guarisce EP1 - Part 2

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In our featured expression che ne so?, che basically stands for "what." We can often translate che ne so as "What do I know?" Sometimes we might translate it as, "How should I know?" It's often a rhetorical question. 

Nilde, ma che mangia il bambino la mattina? -Ma che ne so?

Nilde, but what does the child eat in the morning? -How should I know?

Captions 2-3, Un medico in famiglia Stagione 3 S3 EP1: Ciao famiglia - Part 3

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We've taken care of che. But what about that little word ne? Ne is a particle, called una particella in Italian, and if we look ne up in the dictionary we see it means several things. But mostly, it encompasses both a preposition and the indirect object pronoun "it" or "them." See this lesson about ne.

As mentioned in the lesson, we often don't even notice the word ne because it's so short and because we are not looking for it if we're thinking in English. Once you start thinking in Italian, it will become easier to use and notice. Italians will be very tolerant and understand you anyway, even if you don't use it, so don't worry about it too much. But learning an expression with ne will already make you sound more fluent. 

In our expression, ne means "about it." The tricky thing is that we don't bother with "about it" in English, but in Italian, not always, but in general, we will hear that little ne in there. 

Che ne so? What do I know [about it]?

Finally, we get to so, which is simply the first person singular of the verb sapere (to know).

You might have already learned how to say "I know" and "I don't know" in Italian. Italians add the direct object pronoun lo ("it" or "that").

Sì, lo so (yes I know [that].

Non lo so (I don't know [that]).

But che ne so can also be used in the middle of a sentence, as we would use "I don't know." It's a kind of filler phrase. We can leave it out and the meaning doesn't change much. 

perché, diciamo... -comunque devono sostenere il peso. -Devono sostenere il peso, più che altro devono fare, che ne so, la stessa cosa per un'ora.

because, let's say... -anyway they have to support the weight. -They have to support the weight, more than that, they have to, I don't know, do the same thing for an hour.

Captions 50-51, Francesca Cavalli - Part 2

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Ma tu ti devi aggiornare, sarai rimasto sicuramente, che ne so, ai Pooh.

But you have to get up to date. You must have remained, I don't know, at Pooh.

Caption 66, Un medico in famiglia Stagione 1 EP5 Lele, ti presento Irene - Part 2

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Allora, due colleghi decidono di scambiarsi il posto, firmano un modulo, e se non ci sono problemi, ma gravi, eh, tipo, che ne so, uno deve essere sotto inchiesta.

So, two colleagues decide to switch places, they sign a form, and if there are no problems, but serious huh, like, I don't know, one [of them] has to be under investigation.

Captions 38-40, Il Commissario Manara S1EP8 - Morte di un buttero - Part 12

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Dice, chissà se c'ha un lenzuolo da piegare, se ti manca... che ne so? C'è un tubo che perde acqua...

Saying, who knows if she has some sheets to fold, if you're out of... I don't know... There's a pipe that leaks...

Captions 39-40, Il Commissario Manara S2EP8 - Fuori servizio - Part 1

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For more about particles ci and ne, see Daniela's video lessons (in Italian). 

In this video, Marika explains the particle ne.

Marika explains all about the verb mettere (to put) in this video lesson.

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As you will see, there are dozens of different ways to use mettere. But what can sometimes be tricky is that in English we don’t generally use “put” without some sort of preposition or adverb. We always think of “put in,” “put on,” or “put up,” but in Italian, at least in casual speech, we might hear:

Metti un po’ di sale.
Put [in] a bit of salt.

In a recent episode of Stai lontana da me, there is a discussion between two guys in a couple. One is criticizing the cooking methods of the other:

No, allora cuciniamo per terra

No, so let's cook on the floor

come nel Medioevo. -Ma che c'entra?

like in the Middle Ages. -But what does that have to do with it?

Metti meno olio, no, scusami. -Sì, nella Preistoria.

Put in less oil, right? Excuse me. -Yes, in prehistoric times.

Captions 88-90, Stai lontana da me - Rai Cinema

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The grammatically correct way to say this would be by attaching an indirect pronoun at the end of the verb to mean “in it”:

Mettici meno olio, no?
Put in less oil, can’t you?

If we look carefully, however, we see that earlier in the discussion, they actually do say things the right way:

Chi cucina? Ah, sì,

Who cooks? Ah, yes,

con tutto l'olio che ci metti

with all that oil that you put in,

me l'incrosti da matti, guarda, ogni volta.

you cake them up like crazy, look, every time.

Captions 86-87, Stai lontana da me - Rai Cinema

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In the previous example ci stands for “into them,” where it represents the baking pans. For more on metterci, see this lesson. The lesson also talks about using mettere to say how long something takes—how much time you “put into something” (Marika will talk about this in part 2 of her lesson on mettere).

Here’s another thing to remember with mettere. In an episode of Commissario Manara, there's a dicey situation, and Luca lifts Lara up to help her. She exclaims:

Aiutami. Ah, ah, ah... mettimi giù!

Help me. Ah, ah, ah... put me down!

Captions 40-41, Il Commissario Manara - S1EP2 - Vendemmia tardiva

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In the example above, a single word is formed from the verb and object pronoun together. In this case Lara is using the informal imperative and she’s using herself as the direct object.

In the following example, however, mettimi looks identical, but means something different. This time mi at the end of mettimi is an indirect object and means in this case, “for me.” The direct object is questo (this).

No, mettimi questo sulla scrivania per favore, io vado con la Rubino.

No, put this on the desk for me please. I'm going with Rubino.

Caption 42, Il Commissario Manara - S1EP8 - Morte di un buttero

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And just to add a little something more to the pot, we have the word smettere. An “s” attached to a verb often shifts its meaning to the opposite.

Mi metto al lavoro alle dieci e smetto alle tre di pomeriggio.
I start working at ten and I quit at three in the afternoon.

So a way to ask someone to stop doing something is smettila (stop it)!
In fact, in a recent episode of La Tempesta, Paolo’s neighbor is telling him off.

Terzo, la devi smettere di parcheggiare la Porsche

Third, you have to stop it with parking the Porsche

davanti al pettine delle bici.

in front of the bike rack.

Captions 72-73, La Tempesta - film - Part 1

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Learn more:
Direct object pronouns 
Particelle (little words like mi, ci, ti...)