If you look to the left of this post, click on attractions, there is a decent list of places to see. I would highly recommend the "Fire Mountains". Sit on the side opposite the driver for the best views.
A number of companies are doing full Island tours, but you may be better to split them into a north and a south tour. A more relaxed event.
Driving is easy on the Island and the distances involved are not too far.
My preference is
www.autospluscar.comTaxi's are excellent, most (if not all) speak a decent level of English and will take you directly to where you want to go.
Another option is the local, very cheap bus service. The website lists all services and fares.
http://www.arrecifebus.comFor Island hopping try
http://www.bintercanarias.com , the local Canarian airline. Again all frequencies and prices are on the site. Main routes are to Gran Canaria for a day in Las Palmas, or Tenerife norte for Santa Cruz, La Laguna or Puerto de la Cruz. Regular buses from outside both airport terminals.
http://www.titsa.com for Tenerife
http://www.globalsu.net/index_en.php for Gran Canaria
Enjoy your honeymoon.