Learn Russian: Это пианино бабушки.

To show possession ("grandmother's"), we use the Genitive case. The dictionary form is бабушка. Normally, feminine nouns ending in -а change that to -ы in the Genitive case. However, Russian has a strict spelling rule: never write ы after the letter к. Because of this, the ending becomes -и, giving us бабушки.

The lesson teaches у ... есть for possession. Why isn't it used in this sentence?

The construction у бабушки есть means "grandmother has". You use that to state that someone owns something, like "Grandmother has a piano". In this sentence, we are just describing a specific object ("grandmother's piano"). To link an owner to an object like this, you simply place the owner directly after the object in the Genitive case: пианино бабушки (literally "the piano of grandmother").

Is it possible to change the word order and say Это бабушки пианино?

While people will understand you, it sounds less natural. When using the Genitive case to show possession, the owner almost always comes directly after the thing being owned. Sticking to пианино бабушки is the safest and most standard choice.

Does the word пианино change its ending here?