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Let me swing by tomorrow morning and give you a ride in my car.
Literal
Tomorrow [possessive-の] morning stop-by you [object-を] car [to-に] put-onto-going let's.
Chains three actions together: 立ち寄る ('drop by'), 乗せる ('put on/give a ride to'), and ~ていきましょう ('go, doing — let's'). 乗せていく is a compound with ~ていく signaling the action carries outward toward a destination. The final ~ましょう makes it an offer — 'let me/us do this for you.' 立ち寄る is a compound of 立つ (stand) + 寄る (approach) — literally 'stand-approach.'