Rather then simply wandering about alone, it is a good idea to get a guided tour of the island. The local people will obviously have a much better idea of where the good sites to see are, as well as hopefully knowing how to avoid the rush of other tourists. Here are a few of the major scenes you may want to look at, so be sure to tell your guide you want to visit them.
The Ubud Monkey Park
If wildlife, an in particular monkeys, interest you then you may well want to check out the monkey park in Ubud. You can feed the monkeys small pieces of fruit, purchased at the entrance, and or simply wander around the park. Bear in mind, though, that although they may sometimes act very human they are still animals and you need to be careful. Still, the habitat is extremely well done and the scenery is beautiful.
Pura Luhur Uluwatu
This is a "directional temple" in Bali and is for the purpose of guarding the city from spirits of evil that might come from the south west. This temple is much revered throughout Bali, and it's very important in local rituals and festivals. Founded in the 1100s, it reigns atop a cliff edge and overlooks the Bukit Peninsula. There are also beautiful and ornate temples that were added in the seventeenth century.
The Batur Volcano
This amazing volcano is a must-see in Bali. Still active, this volcano is one of the most popular attractions in Bali. The caldera alone is over 10 kilometers wide and approximately 450 meters deep. Much legend and lore surrounds the volcano, and it is said to be Goddess Danu's throne.
Barong Dance
The Barong dance is generally used to mean a dance where animals are represented by masks of the dancers, so for example in the Barong Macan dance a tiger mask is used (Macan means Tiger). In fact, the original meaning of the word Barong was a character in Balinese mythology. He is the king of spirits, and leader of the hosts of good. Although represented in different regions by different animals, the most common representation is of him as a lion, representing strength.
No doubt you will probably find your own favorite place to go in Bali, but remember that there are lots of people out there who have jobs taking advantage of tourists! Be careful and be informed before you go and you will have no nasty surprises. Turn down free merchandise and services--they will expect something in return.