Here are some things I have found out that make life easier
The local bus company called Titsa have a web site in English http://www.titsa.com/index.php?idioma=1. I found this very helpful, go into the google section which says “Find here the best route for reach your destination and planned schedules”. This will plan your route from start to finish.
There is also a bus number 111 that leaves from Las Americas bus station and takes you to the airport, it is every 30 minutes, very convenient only takes about 30 minutes and costs around 3 euros. Much better value than a taxi which could cost you 25 to 35 euros.
You can purchase a card called a bono card at Spanish Style or Holiday Time shops, or bus stations, for 15 euros, this gives you a discount on travel and is valid till the 15 euros are spent.
This card also can give you your second journey for 0.10 cents if you travel within 2 hours and use a different number bus, than the first journey.