詩 is a more general and common word for 'poem' or 'poetry', while 詩歌 often implies a broader or classical scope, especially including both Japanese and Chinese traditions.
和歌 refers specifically to classical Japanese poetry (tanka), whereas 詩歌 can encompass both Japanese and Chinese poetry.
Etymology
Compound of 詩 (poetry, especially Chinese-style) and 歌 (song, Japanese-style poetry). The reading しいか is the standard Sino-Japanese on'yomi; しか is an older or alternative reading.
