Learn Russian: Мы надеемся на хорошую погоду.

In Russian, verbs often pair with specific prepositions. The verb надеяться (to hope) always teams up with the preposition на when expressing what you are hoping for. You cannot use words like для or за here.

Why is the phrase хорошую погоду in the accusative case?

The preposition на takes the accusative case when it indicates the target of your hopes. Since погода (weather) is a feminine noun, its ending changes to -у, and the feminine adjective хорошая (good) changes to match it with the -ую ending.

Надеемся translates to 'we hope', but isn't that a reflexive verb ending in -ся?

Yes! The infinitive is надеяться. In Russian, many verbs have the reflexive -ся or -сь ending. While this often means doing an action to yourself, some verbs like надеяться just have this ending naturally. When conjugated for мы (we), it becomes надеемся.

How could we connect this sentence to a real conditional 'Если ... то ...' (If ... then ...) clause?

This sentence is a great setup for a conditional! You could follow it up by saying: Если погода будет хорошая, то мы пойдём в парк (If the weather is good, then we will go to the park). Notice that in the conditional part, хорошая погода is back in the nominative (dictionary) case because it acts as the subject of that new clause.