How do you use the phrase 留宿 （liu2su4） （i.e. “To stay over at someone's house")
Yeah it's kind of formal and "old-fashioned".

Expressing 今天我打算在李雷家留宿 more naturally can be 今天我打算住李雷家 or 今晚我打算住李雷家.

今天 in this sentence implicitly means 今晚 (tonight).