Lucia diligenter studet.

• Lucia is the subject: the person doing the action.
• diligenter is an adverb: it tells us how she studies.
• studet is the verb: she studies.

So the structure is basically:

• Lucia = subject
• diligenter = adverb
• studet = verb

In Latin, verb endings often include the subject.
The ending -t in studet tells you it is third person singular, so it means:

• he studies
• she studies
• it studies

Because the subject here is Lucia, we understand it as she studies.

Latin usually does not need a separate subject pronoun if the verb already shows the person and number.

So instead of saying something like Lucia she studies diligently, Latin simply says:

• Lucia diligenter studet

The verb ending -t already tells us the subject is he/she/it.
If Latin adds a pronoun like ea (she), it is usually for emphasis or contrast.

Lucia is in the nominative singular. That is the case normally used for the subject of the sentence.

A learner might notice that Lucia also looks like a typical first-declension noun ending in -a. That is correct: as a name, it follows the same general pattern. In this sentence, it is the subject, so the nominative form Lucia is exactly what we expect.

diligenter is an adverb, meaning diligently, carefully, or with care.

It is related to the adjective diligens / diligent-, meaning careful, diligent, hard-working.
A common Latin way to form an adverb from a third-declension adjective is to use -ter, which is why we get:

• diligens = diligent, careful
• diligenter = diligently, carefully

So diligenter tells us the manner of the action.

Latin word order is more flexible than English word order because Latin uses endings to show grammatical relationships.

So Lucia diligenter studet is natural, but other orders are also possible, for example:

• Lucia studet diligenter
• Diligenter Lucia studet
• Studet Lucia diligenter

These can all mean essentially the same thing, though the emphasis may change.
The version with the verb at the end, Lucia diligenter studet, is a very normal and common Latin arrangement.

Yes. The verb studeo, studēre, studui can mean more than simple classroom-style study.

Depending on context, it can mean:

• to be eager for
• to devote oneself to
• to apply oneself to
• to study

So in a simple beginner sentence like this, studet is usually best understood as studies, but the verb has a somewhat broader sense of being earnestly engaged in something.

Often, no—not in the same way English does.

Latin studeo commonly takes the dative case for the thing a person is devoted to or eager for. For example, Latin may say something more like:

• She is devoted to literature rather than
• She studies literature

So a learner should be aware that studeo does not always behave like English study. In Lucia diligenter studet, there is no object, so this issue does not appear directly, but it is still useful to know.

Studet is present tense.

That means the sentence describes something happening in the present, such as:

• Lucia studies diligently
• Lucia is studying diligently

Latin present tense can cover both a simple present and, depending on context, something like an English present progressive.

Studet can be parsed as:

• verb
• present
• active
• indicative
• third person singular

From the dictionary form studeo, studēre.

That full parsing tells us it is a straightforward statement: she/he is studying or she/he studies.

A common classroom pronunciation would be roughly:

• LUC-i-a di-li-GEN-ter STU-det

More carefully:

• Lucia: the stress is usually on Lu-
• diligenter: the stress falls on -gen-
• studet: the stress falls on stu-

If you are using a restored classical pronunciation, c in Lucia is pronounced as k, not like English sh or s. So Lucia sounds more like Loo-kee-a than modern Italian-style Loo-chee-a.

Latin often places the verb near or at the end of the sentence. This is a very common stylistic pattern, especially in textbook Latin and many classical sentences.

So Lucia diligenter studet feels very natural in Latin, even though English more often prefers:

• Lucia studies diligently

Latin does not rely mainly on word order to show grammar, so ending with the verb is perfectly normal.