Learn Hardcore Italian: Trovo un codice nel libro rosso.

Trovo is the first-person singular present indicative of trovare (“to find”). In English you say “I find,” so in Italian you use trovo (“I find”). If you wanted the infinitive, you’d keep trovare (“to find”), but the sentence would need a different structure (“Voglio trovare un codice…” etc.).

Why is there an indefinite article un before codice? Could I use il or omit the article?

Un is the indefinite article meaning “a” or “an,” indicating you’re talking about any code, not a specific one.

If you know exactly which code you’re referring to, use the definite article: il codice (“the code”).
In Italian you generally can’t omit the article with singular countable nouns, so you wouldn’t say just codice here.

What does nel stand for? It isn’t just in, right?

Nel is a contraction of in + il.

In is the preposition “in.”
Il is the masculine singular definite article.
Put together, in + il libro becomes nel libro (“in the book”).

Why is the adjective rosso placed after the noun libro, unlike English where adjectives come first?

In Italian, most adjectives—especially color adjectives—come after the noun they modify. Saying libro rosso (“book red”) is the normal pattern. Placing the adjective before (e.g. rosso libro) is unusual and often changes the nuance or sounds poetic.

How do I know the gender of codice and libro, since they end in -e and -o?

Libro ends in -o, which is almost always masculine.
Codice ends in -e; Italian nouns in -e can be masculine or feminine, so you must learn them case by case. In this instance, both codice and libro are masculine, as shown by the articles un and il (in nel).

How would I express “I found a code in the red book” (past tense) instead of “I find”?

You’d switch to the passato prossimo of trovare. The sentence becomes:
Ho trovato un codice nel libro rosso.
Here ho is the present of avere, and trovato is the past participle of trovare.

Could I use a different preposition instead of in here, like su or dentro?

In is the standard preposition for something located inside a container (the book).
Dentro can mean “inside,” but you’d normally say dentro al libro (which sounds heavier and less idiomatic for this context).
Su means “on” or “on top of,” which changes the meaning (“on the book”), so it wouldn’t fit if you want “inside the red book.”