Learn Italian: Lui verrà in palestra per allenarsi. - He will come to the gym to train.

Verrà is the "he/she/it" future tense form of venire (to come). Venire is one of a special group of verbs that drop a vowel in their future tense stem, leaving two Rs right next to each other. Instead of following the regular pattern to become venirà, it squeezes together into verrà.

Why do we use in instead of a for "to the gym"?

In Italian, certain common locations always take the preposition in without any article like "the". In palestra (to/at the gym) is a set phrase you just have to memorize, very much like in banca (to the bank) or in biblioteca (to the library).

Why is per used before the verb here?

When you want to express the purpose of an action, translating to "in order to" in English, Italian uses per followed by a verb in its infinitive (dictionary) form. Here, per allenarsi literally means "in order to train himself".

How is the word allenarsi formed?

It is a combination of the verb allenare (to train) and the reflexive pronoun si (himself). Whenever you need to attach a pronoun to an infinitive verb, you drop the final -e of the verb and attach the pronoun directly to the end: allenare + si = allenarsi.