Learn Polish: Niosę ciężki koszyk. - I am carrying a heavy basket.

Niosę comes from nieść, which is a unidirectional verb. This means you are carrying the basket right now in one specific direction, like walking straight home. If you were talking about carrying it regularly as a habit, or walking around aimlessly with it, you would use the multidirectional verb noszę.

How would I say 'I will bring a heavy basket' using the verbs from this lesson?

You would use the perfective verb przynieść, which means to successfully carry something to a destination (to bring). The 'I' form is przyniosę, so the sentence becomes Przyniosę ciężki koszyk.

Is ciężki koszyk in the accusative case here? Why doesn't the ending change?

Yes, it is in the accusative case because it is the direct object of the action. However, koszyk (basket) is a masculine inanimate noun. In Polish, the accusative form for masculine inanimate nouns and their adjectives looks exactly the same as their basic dictionary form (the nominative case).

Why does the verb nieść change to niosę with an 'o' here?

The verb nieść has a vowel shift in its present tense conjugation. The 'e' changes to an 'o' for the 'I' form (niosę) and the 'they' form (niosą). For the other forms, the 'e' stays, for example: you carry is niesiesz.