Learn Russian: Я даю газету. - I am giving the newspaper.

Because it is the direct object of the sentence (the thing being given), so it must be in the accusative case. For feminine nouns ending in -а like газета, the ending changes to -у in the accusative case.

Wait, I thought inanimate objects don't change their endings in the accusative case?

That rule is only for masculine and neuter nouns! Feminine nouns ending in -а or -я always change their endings in the accusative case, even if they are inanimate objects like a newspaper.

The dictionary form of the verb is давать (to give). Why is the 'I' form даю and not даваю?

The verb давать is a bit tricky! In the present tense, you must drop the -ва- part from the middle of the word before adding the normal endings. So, instead of даваю, it correctly becomes даю.

Does я даю mean 'I give' or 'I am giving'?

It means both! Russian only has one present tense. Depending on the situation, я даю can mean you are giving the newspaper right at this moment, or that you give it regularly.