Ita discipuli discunt et quid prosit et quid obsit, et cur desidia vitanda, perseverantia autem laudanda sit.

Ita means thus, so, or in this way. It usually points back to something in the previous context: this is how the students learn.

So Ita discipuli discunt... means Thus the students learn... or In this way the students learn...

The main clause is:

Ita discipuli discunt = Thus the students learn

What they learn is then expressed by a series of indirect questions:

• quid prosit = what is beneficial / what helps
• quid obsit = what is harmful / what hinders
• cur desidia vitanda, perseverantia autem laudanda sit = why laziness should be avoided, while perseverance should be praised

So the sentence is basically:

Thus the students learn both what helps and what harms, and why laziness should be avoided but perseverance praised.

The first two ets form a pair:

et ... et ... = both ... and ...

So:

et quid prosit et quid obsit = both what helps and what harms

Then the third et adds another item to discunt:

• they learn what helps
• they learn what harms
• and they learn why laziness should be avoided and perseverance praised

So the third et is just a normal and.

Here quid means what and introduces an indirect question:

• quid prosit = what is beneficial
• quid obsit = what is harmful

A learner might expect quod because English sometimes uses that which, but Latin commonly uses quid in indirect questions after verbs like discunt.

So this is not a relative clause like that which helps. It is an indirect question: what helps?

They are subjunctive because they are in indirect questions.

After a verb like discunt (they learn), Latin often uses indirect-question clauses introduced by words like:

• quid = what
• cur = why

And the verb in an indirect question goes into the subjunctive:

• quid prosit
• quid obsit
• cur ... sit

This is one of the most important regular uses of the Latin subjunctive.

They are forms of the irregular compounds:

• prosum, prodesse = to be useful, help, benefit
• obsum, obesse = to be harmful, hinder, hurt

So:

• prosit = it may be useful / it is useful in an indirect-question subjunctive
• obsit = it may be harmful / it is harmful in an indirect-question subjunctive

Because the sentence already gives the general idea, these can be understood broadly as what is helpful and what is harmful.

Yes. Prosum and obsum normally take the dative of the person or thing affected:

• hoc discipulis prodest = this helps the students
• hoc discipulis obest = this harms the students

In this sentence, the dative is not stated because the idea is general. Latin can leave it understood when the context makes it obvious. The general sense is something like what is beneficial and what is harmful—implicitly, beneficial or harmful for a person / for life / for conduct.

They are gerundives:

• vitanda from vitare = to avoid
• laudanda from laudare = to praise

With a form of esse, the gerundive often expresses necessity, appropriateness, or duty:

• desidia vitanda sit = laziness should be avoided / must be avoided
• perseverantia laudanda sit = perseverance should be praised / deserves to be praised

So this is the gerundive of obligation or a closely related use.

Both are nominative singular feminine:

• desidia = laziness
• perseverantia = perseverance

They are the subjects of the implied or expressed sit:

• desidia vitanda [sit]
• perseverantia laudanda sit

The gerundives agree with them:

• vitanda matches desidia
• laudanda matches perseverantia

All are feminine singular nominative.

Latin often leaves out repeated forms of esse when the structure is parallel and the meaning is clear.

So the sentence really means:

• cur desidia vitanda sit
• [et] perseverantia autem laudanda sit

Or, more fully in sense:

• why laziness should be avoided
• and why perseverance, on the other hand, should be praised

The first sit is omitted because the second one makes it clear.

Autem means however, but, or on the other hand. Here it marks a contrast between:

• desidia = laziness
• perseverantia = perseverance

So the sense is: laziness should be avoided, perseverance however should be praised.

Also, autem is a postpositive word in Latin. That means it usually comes second in its clause or phrase, not first. So Latin says:

perseverantia autem

not normally autem perseverantia.

To an English speaker, yes—but it is very normal Latin.

A few things to notice:

• discipuli comes before discunt: subject before verb
• the indirect questions follow the main verb
• sit comes at the end, which is very common in Latin
• the contrastive word autem comes second in its phrase

Latin word order is much freer than English word order because grammatical endings carry so much of the meaning. So the sentence is arranged less by fixed position and more by emphasis and style.

Yes. More literally it is:

what may help and what may harm

or

what is beneficial and what is harmful

Because these are indirect questions in the subjunctive, English often uses a plain present translation. The important point is not that the action is uncertain, but that Latin uses the subjunctive after quid and cur in indirect questions.