From the Santa Marta Airport, you can take a bus or a taxi to Santa Marta center. You can also take a taxi that will directly get you to Minca, but you won't be able to go further as only all-road vehicles can reach Mano Verde.
You can either take a 'colectivo' minibus from the central market, a mototaxi or a taxi to reach Minca from Santa Marta. It will take you between 40 min and an hour to get to Minca and you will pay between 9000 COP and 70000 COP depending on the chosen vehicle.
Once in Minca, you will have to find the mototaxis at the taxi stations. They know the path and are used to drive with big backpacks strapped to their moto. It will take about 40 min and will cost you 40000 COP. You might also be able to find a 4x4 car to come here, hosting up to 5 people and costing around 150000 COP