うれしい" is an emotion that is happy momentarily. You can say it even when it is continuing to a certain extent. Example 〇プロポーズされてうれしい。 (I am happy to propose to him.) ○プレゼントをもらってうれしい。(I am happy to receive a present.) ○テストが100てんでうれしい。( I am happy to received a 100 on the test.) "しあわせ" is used when you are satisfied with the current state. Example ○けっこんせいかつは しあわせです。(His married life is happy.) ○すきなしごとができて しあわせです。(I am happy that I can do my favorite job.)

嬉しい describes short term happiness where as 幸せ is more long term. You'd be 嬉しい if someone surprised you with a gift and would feel 幸せ that you have such a nice friend. You'll hear 嬉しい more in your daily life than 幸せ.