I want to spend the rest of my life pursuing my ideals.
Literal
Rest-of-life [topic-は] own [possessive-の] ideals [possessive-の] pursuit [on-に] spend-want.
余生 ('the rest of one's life / later years / remaining years'). 追求 ('pursuit'), used for abstract goals, ideals, truth. 費やす ('spend / expend'), here for time spent in pursuit. A life-aspiration statement.