Le professeur universitaire leur conseille de garder les brochures et les billets comme souvenirs de ce festival.

In French, conseiller à quelqu’un quelque chose means to advise someone something.

• leur is an indirect object pronoun = to them (à eux / à elles).
• les is a direct object pronoun = them (as the direct object).

In this sentence, the professor is giving advice to them, not advising them as a direct object. So you need the indirect form leur, not les.

• leur (without s) can be:
  • an indirect object pronoun = to them (this is the case here).
• leurs (with s) is a possessive adjective = their (plural thing possessed).

Here, leur is not showing possession; it is a pronoun replacing à eux / à elles. So you use leur, not leurs.

Object pronouns (like me, te, lui, nous, vous, leur, le, la, les) normally go before the conjugated verb:

• Le professeur leur conseille de garder...

You cannot place leur after conseille in this structure.
Only in affirmative imperatives can many pronouns go after the verb (for example: Conseille‑leur de garder...).

Many French verbs are followed by de + infinitive.
Conseiller is one of them in the pattern:

• conseiller à quelqu’un de faire quelque chose
  → Le professeur leur conseille de garder...

You must include de before the infinitive garder.
conseiller garder (without de) is incorrect in standard French.

The full underlying pattern is:

• conseiller à quelqu’un de faire quelque chose
  
  • conseiller = to advise
  • à quelqu’un = to someone → replaced by leur
  • de faire quelque chose = to do something → here de garder les brochures et les billets

So the sentence follows this model exactly:
Le professeur universitaire (subject) leur (indirect object pronoun) conseille (verb) de garder les brochures et les billets (infinitive clause).

• garder is the most neutral, common verb meaning to keep / to hold on to something you already have.
• conserver is a bit more formal and often implies preserving something carefully (food, archives, heritage).
• tenir usually means to hold in your hand or to keep in a certain state, and would not be used for keeping brochures as souvenirs.

For everyday speech about keeping objects, garder is the natural choice.

Using les (the definite article) suggests that:

• the brochures and tickets are specific, already known items (for example, the ones they just received at this festival).

des brochures et des billets would introduce them as some brochures and tickets, more indefinite or vague.

Because the context is specific (the ones from this festival), les is appropriate.

souvenirs is in the plural because:

• the brochures and the tickets together create multiple memories / mementos of the festival.
• It’s very common to talk about des souvenirs in the plural when referring to physical keepsakes.

You could say comme souvenir (singular), but that would emphasize the idea of one overall remembrance rather than multiple keepsakes. Plural fits more naturally with several physical objects.

In this context:

• comme means as / in the role of.
  
  • les garder comme souvenirs = keep them as souvenirs.
• en before a noun usually means in / made of / covered in (for example, une sculpture en bois = a sculpture in wood).

So comme souvenirs is the correct way to express as souvenirs.
en souvenirs would sound wrong here and would not mean the same thing.

Both can refer to a university professor, but they are not identical:

• professeur d’université explicitly means professor at a university.
• professeur universitaire uses universitaire as an adjective meaning academic or related to higher education. It sounds a bit more formal or descriptive (an academic professor).

In everyday speech, professeur d’université is more common.
professeur universitaire is correct and understandable, just slightly more formal or stylistic.

In French, most descriptive adjectives (especially those of origin, relationship, or classification) come after the noun:

• un professeur universitaire
• un professeur médical, un professeur musical (same pattern)

Only certain common, short adjectives (like grand, petit, vieux, bon, mauvais, beau) often come before the noun.
Since universitaire is a classifying adjective, it naturally follows professeur.

The choice between ce and cet depends on the sound that follows:

• ce is used before a masculine noun beginning with a consonant sound:
  
  • ce festival, ce professeur, ce livre
• cet is used before a masculine noun beginning with a vowel or mute h:
  
  • cet étudiant, cet homme

festival starts with an f sound (a consonant), so the correct form is ce festival.

In French, conseiller is usually followed by an infinitive in this type of structure:

• conseiller à quelqu’un de faire quelque chose

The structure conseiller que + subjonctif is rare and tends to sound foreign or very formal/archaic.
In normal modern French, you should say:

• Le professeur universitaire leur conseille de garder les brochures et les billets...

Using qu’ils gardent here would not be natural.