I Need To Travel To Panama City From Buenos Aires or Santiago Chile?

It will take you 8-9 days if you don't stop and about $500-600 (including food and lodging). You can go by bus to Turbo, Colombia (via Medellin) and then take a boat to the frontier town fo Capurgna and then another boat to Puerto Obaldia Panama from there you can take a boat to Colon (they are infrequent so it might be better to fly for $80-$90).