Quick answer
tomar = to take hold of, to take in — a drink, a bus, a decision, the sun.
llevar = to take something or someone with you, away from here.
If the thing moves along with you to another place, it is llevar. If you simply pick it up or consume it, it is tomar.
Side by side
Memory trick
Ask whether the object travels with you. Travelling with you means llevar; everything you drink, board or decide is tomar.
Typical mistakes
Voy a tomar a mi hermana al aeropuerto.
Voy a llevar a mi hermana al aeropuerto.
Your sister moves with you to another place, so it is llevar. Tomar here sounds like you are grabbing her.
Llevo un café cada mañana.
Tomo un café cada mañana.
Drinking something is tomar. <i>Llevo un café</i> means you are carrying a coffee around.
Check yourself
Fill in: "¿Me ___ al centro en coche?"