Learn Finnish: Ostin kaupasta uuden kihartimen ja teroittimen. - I bought a new curling iron and a sharpener from the store.

They are derived from the verbs kihartaa (to curl) and teroittaa (to sharpen) using the instrument suffix -in. This suffix creates a noun meaning a tool for doing an action, giving us kiharrin (curling iron) and teroitin (sharpener). When we add a case ending to these nouns, their stem changes to -ime-, which is why we see kihartime- and teroittime- here.

Why do the consonants change between the basic dictionary forms (kiharrin, teroitin) and the forms in this sentence (kihartimen, teroittimen)?

This is standard consonant gradation! The nominative forms kiharrin and teroitin end in a consonant, meaning their final syllables are closed. This forces the weak consonant grades (rr and a single t). When the stem changes to -ime- to attach the -n object ending, the syllables open up (-ti-), triggering the strong grades (rt and double tt).

Why are the nouns in the accusative case (ending in -n) instead of the partitive?

Because the verb ostaa (to buy) expresses a completed action here. You successfully bought and received the entire curling iron and sharpener, making them total objects. Total objects in an affirmative, active sentence take the genitive-like accusative -n ending.

Does the adjective uuden apply to both items in this sentence?