Stella, you are getting lots of very good suggestions here, both from visitors and residents. Although this forum is called the Bali Expat forum, the truth is that the travel section of this forum is likely the best and most balanced travel forum on the net.
Chad from Denver has provided some very enthusiastic and in my opinion excellent tips. His exuberance for Bali, as well as his willingness to share in the hopes that your experiences here are as delightful as was his own, is self-evident.
I also agree with “Guest’s” last post concerning the use of a Balinese driver during portions of your stay. The advantages of using a good local driver far exceed the nominal additional costs of renting a “Jimmy” and self driving...especially on your first visit to Bali. If you need some recommendations, I can offer two members of my family, Agung Blado, or another, Wayan Dedik who is university degreed in the US and has worked as a consultant and translator for the BBC and other foreign firms needing a high level of expertise, skill and knowledge of Balinese culture.
In Bali, the best place for snorkeling or diving, outside of a remote area in northwest Bali, a part of the National park, is Amed. In Amed, I cannot suggest anyone better than Eco Dive. You can check their web site at
http://www.ecodivebali.com and see for yourself. A very good friend of mine, John Huxley runs this first rate operation, and all the folks I have sent his way....be they first timers or experienced divers, have been delighted that I steered them his way.
As for “must see” in Ubud, and considering your desire for culture, there are several great museums of art....BUT, if you really want to immerse yourself in culture, again, the use of a local driver is almost essential. Either Agung, or Dedik could easily get you into a local ceremony, not some dance and performance for tourists as you will find in and around the royal palace, but the “real deal” where you will likely be among a small handful of other “tamu” (or guests). An all night long Calonarang at at Pura Dalem would be the best you could hope for, and every guest I have taken to one of these has left Bali forever changed. Very likely, given the timing of your visit, would also be a good local banjar cremation, which is a day time, mostly afternoon, experience.
The Balinese always welcome foreign visitors to their ceremonies so long as they are properly dressed and polite.
Also essential to get a grasp on local culture is to visit the many villages that surround Ubud. It is here that you will find the compounds of Balinese families that are the artists, and the core of the cultural heart of Bali. Most of these compounds do not hang a sign announcing their trade, nor will you see tourist buses lined up there, as you will in the central market of Ubud. Once again, a knowledgeable Balinese local is essential for this experience.
So, and once again, have a great time. Bali is what you make of it, or more correctly, what you allow it to make of you.