Learn Finnish: Miten suhtaudut tähän uuteen haasteeseen? - How do you react to this new challenge?

This is due to the verb suhtautua. It is a verb of reacting or having an attitude towards something, and it strictly requires its target to be in the illative case (the "into" case). Even though we say "react to" in English, in Finnish you take an attitude "into" a challenge.

The English translation says "react". Could I just use the verb reagoida here?

You could, and it would also take the illative case: Miten reagoit tähän uuteen haasteeseen? However, there is a nuance difference. Suhtautua implies a broader, more continuous stance or attitude towards the challenge. Reagoida often refers to a more immediate, spontaneous, or single reaction.

Can you remind me why the adjective uusi becomes uuteen here?

The word uusi belongs to a group of older Finnish words ending in -si. When building cases, this -si generally changes to -te- to form the stem. The stem is uute-, and when you add the illative ending -en (matching the preceding vowel + n), you get uuteen.

Why does haaste take the illative ending -seen instead of just doubling the vowel and adding -n?