La salumiera consiglia il prosciutto più fresco, ma la fornaia preferisce prendere del formaggio.

They are job nouns.

• la salumiera = the female deli worker / delicatessen shopkeeper / woman who sells cured meats
• la fornaia = the female baker

The masculine forms are:

• il salumiere
• il fornaio

So the sentence is talking about two women.

Because Italian usually marks the gender of many profession nouns.

Here:

• salumiere → masculine
• salumiera → feminine
• fornaio → masculine
• fornaia → feminine

The article also changes to match:

• la salumiera
• la fornaia

This is one of the first things English speakers notice, because English job titles usually do not change form this way.

Because in this sentence the professions are being used as full nouns meaning the deli woman and the baker.

Italian often uses the definite article with a profession when you are talking about a specific person as the subject of a sentence:

• La fornaia preferisce...
• Il medico arriva...

But after essere, Italian often drops the article when simply stating someone’s job:

• È fornaia. = She is a baker.
• È salumiera. = She is a deli worker.

So in your sentence, la is perfectly natural because we mean the baker and the deli worker, not just she is a baker.

Both are in the present indicative, third-person singular.

• consiglia = he/she recommends
• preferisce = he/she prefers

They are singular because each verb has a singular subject:

• La salumiera consiglia...
• la fornaia preferisce...

Because Italian, like English, often uses preferire + infinitive when the same person does both actions.

• La fornaia preferisce prendere del formaggio.
• Literally: The baker prefers to take some cheese.

The subject of preferisce and prendere is the same person: la fornaia.

If the subject changed, Italian would normally use a different structure, often che + subjunctive:

• La fornaia preferisce che io prenda il formaggio.
• The baker prefers that I take the cheese.

Usually it means the freshest ham.

In Italian, definite article + più + adjective often forms a relative superlative:

• il più fresco = the freshest

So:

• il prosciutto più fresco = the freshest ham

However, depending on context, English might sometimes phrase it as the fresher ham, especially if only two hams are being compared. But the most standard way to understand it here is the freshest ham.

Because in Italian, descriptive adjectives often come after the noun.

So:

• prosciutto fresco
• formaggio buono
• pane caldo

That is the normal pattern.

Putting the adjective before the noun is sometimes possible, but it can change the tone or emphasis. In a straightforward phrase like this, prosciutto più fresco is the natural order.

Because the adjective has to agree with prosciutto.

• prosciutto is masculine singular
• so the adjective must also be masculine singular
• therefore: fresco

Compare:

• il prosciutto fresco = masculine singular
• la carne fresca = feminine singular
• i salumi freschi = masculine plural
• le focacce fresche = feminine plural

The same agreement still applies in più fresco.

Del formaggio means some cheese.

Here del is a partitive article, used for an unspecified amount of something, especially with uncountable nouns like food.

• del formaggio = some cheese
• del pane = some bread
• del vino = some wine

It is formed from:

• di + il = del

Why not il formaggio?

• il formaggio would mean the cheese, a specific cheese already identified
• del formaggio means an unspecified quantity: some cheese

Why not just formaggio with no article?
Sometimes Italian can omit the article in certain expressions, but prendere del formaggio is the normal, natural way to say to get some cheese.

Because the two nouns are being used differently.

• il prosciutto più fresco refers to a specific item or selection: the freshest ham
• del formaggio refers to an unspecified amount: some cheese

So the articles show different meanings:

• il = the
• del = some

This difference is very important in Italian, especially with food.

In this sentence, recommends is probably the best choice.

• consigliare can mean to advise, to suggest, or to recommend
• when the object is a product or thing, recommend is often the most natural English translation

So:

• La salumiera consiglia il prosciutto più fresco
  = The deli worker recommends the freshest ham.

If the sentence were about advising a person more generally, advise might fit better.

Yes, because preferisce prendere del formaggio emphasizes the action: she prefers to get / buy / take some cheese.

If you said:

• La fornaia preferisce il formaggio

that would usually mean the baker prefers cheese, more generally, as a preference between things.

But in your sentence, the idea is about what she chooses to take, so preferisce prendere is a very natural structure.