Learn Portuguese (Portugal): Quero alugar uma bicicleta na loja.

In Portuguese, when you use a verb like querer (to want) followed by another action, the second verb stays in its base form, called the infinitive. The infinitive alugar already means "to rent", so you do not need an extra word for "to". You just put them directly together: Quero alugar.

If I wanted to say "He wants to rent", does the word alugar change?

No, only the first verb (querer) changes to match the person! You would change Quero to Ele quer (He wants), but alugar stays exactly the same in its base form. So, it would be Ele quer alugar.

Where is the word for "I" in this sentence?

In Portuguese, you can usually drop subject pronouns like eu (I) because the verb ending already tells you who is doing the action. Quero specifically means "I want", so saying Eu quero is optional and usually left out in everyday speech.

Why does the sentence say na loja for "at the shop" instead of using em?

na is a mandatory combination (contraction) of the preposition em (in/at/on) and the feminine word for "the", which is a. Because loja (shop) is a feminine noun, "at the shop" becomes em + a = na loja.