Learn Finnish: Suuni on pieni. - My mouth is small.

The basic dictionary form of mouth is suu. To say my mouth, we attach the first-person possessive suffix -ni directly to the end, giving us suuni.

Do I need to use the word minun (my) before suuni?

In standard written Finnish, you can just say suuni to mean my mouth. Adding minun (like minun suuni) is also correct and adds emphasis, but the -ni suffix already contains the meaning of my. In everyday spoken Finnish, people often just say mun suu without the suffix.

The word pieni also ends in -ni. Is that a possessive suffix too?

No, pieni is just the basic dictionary word for small. The letters ni at the end are simply part of the root word, not a possessive suffix!

Does adding -ni change the way the vowels in suu are pronounced?

No, the pronunciation of the base word suu (a long U sound) stays exactly the same. You just add the -ni syllable right after it as a single continuous word.