Learn Hardcore Finnish: Sovitan uusia kenkiä kaupassa.

Sovitan is the 1st person singular present tense of the verb sovittaa (type 1), meaning to try on. The stem is sovitta- and the personal ending for I is -n.

Why is uusia kenkiä in the partitive plural instead of the nominative or the accusative plural?

Finnish uses the partitive case for
• the object of an action that is ongoing or incomplete
• indefinite quantities, especially in plurals
Here uusia kenkiä is partitive plural because the action of trying on shoes is unfinished and you are not specifying particular fully affected shoes.

Why is the adjective uusia also in the partitive plural?

In Finnish adjectives agree with the nouns they modify in case and number. Since kenkiä is partitive plural, uusi becomes partitive plural uusia.

What case is kaupassa, and why is it used here instead of something like kauppaan?

Kaupassa is the inessive case with suffix -ssa, indicating location inside. It means in the shop.
Use the inessive for in or on a location. Kauppaan is the allative case meaning into the shop.