Learn Finnish: Minä aion levätä.

This sentence uses a verb chain. When you use a verb expressing a plan or intention, like aikoa (to intend / be going to), the second verb stays in its basic infinitive form. Just like in English, we say "I intend to rest".

The translation says "I am going to", but does Finnish have a future tense?

Finnish actually doesn't have a dedicated future tense! To talk about the future, you usually just use the present tense. However, using aikoa (to intend / plan to) is a very common way to specifically show that you are planning to do something in the future.

The current lesson teaches us to use minun (my). Why does this sentence use minä (I)?

The verb aikoa takes a normal subject (minä), just like most verbs. You only use the genitive form (minun) with specific obligation verbs like täytyy (must). So you say minä aion (I am going to) but minun täytyy (I have to).

What is the dictionary form of aion?

The basic form is aikoa. Notice that the k disappears when it is conjugated to the "I" form (aion). This happens because of consonant gradation, a common pattern in Finnish where certain consonants weaken or disappear.