Since this is the OP's honeymoon, and he's a lawyer in New York who just came off a six-month trial, and he already told us he's staying at amanjiwo and in "jimbaran and ubud," I'm going to hazard a guess that in Bali the OP is either splitting time between the Four Seasons Jimbaran Bay and Sayan or the VSOE Jimbaran Puri and Ubub Hanging Gardens. In either event, I'd say that the resorts are certainly nice enough to justify spending whole days just relaxing at the resort-- especially if one is on one's honeymoon. (I say this as one who just came back from my third visit to Bali in three years, all with Mr. Megatop, this time staying at FS Sayan. I haven't written a proper trip report yet.) The FS Sayan, for example, has lots of activities available daily, from yoga to biking to trekking and even a marriage-solemnizing ceremony.

While many of the resorts in Bali are spectacular, Bali is a magical place in general and one really must get outside the resorts to experience and enjoy it properly. I'd suggest hiring a driver such as Dewa Marco, who is an FT member and has a blog; there's also a recent thread in this forum about drivers in Bali. Bali is a huge place that takes some time to get around, so I'd suggest tying your sightseeing to where you're staying. There's not a huge amount of sightseeing near Jimbaran, although seeing the Uluatu temple and some traditional Balinese dancing at sunset is a must-do; just beware of theiving monkeys. Seriously. Kuta and Denpasar are completely skippable. Ubud and the surrounding villages are centers for the arts, all manner of arts. I'd just ask your driver to show you around different places where you can find good examples of the various crafts: silversmithing, woodcarving, stonecarving, painting, whatever. The areas outside Ubud and going up toward the volcanoes have some spectacular temples your driver can take you to. In Ubud, Mozaic restaurant is absolutely worth it (Mr. Megatop and I went twice on our last trip); Ku De Ta in Seminyak is not. Bali is a very big place, probably bigger than you are expecting. Do not hesitate to hire a driver for four or five days. But do be sure to get plenty of time to just stay in the resorts and enjoy them, and your prescribed honeymoon activities.

Can't speak to amanjiwo. Yet. Headed there in December. A regular poster in the Luxury Hotels forum, Ericka, did a lengthy detailed Trip Report on amanjiwo (and the Ritz in Singapore) that's worth reading. My own plans are to take advantage of most of the offerings on the "Cultural Trails" package offered by amanjiwo.

I've posted "What to do in Singapore in Four Days" in a thread or two in this forum and the Singapore forum within the last couple of months. Do the cultural things and experience the city, maybe even visit a tailor. (Prices will shock a New Yorker. Try Jack's Custom Tailor.) But do not miss the food-- that is one of Singapore's strongest points. For fusion, I like the restaurant at the New Majestic. For excellent Sichuan, try the Hometown restaurant in Smith Street, Chinatown. For high-end Cantonese, try Jade at the Fullerton Hotel. My mom recently went to the Courtyard restaurant at the Raffles Hotel and couldn't say enough good things about it, especially the setting. My mom also liked the Chinese restaurant at the Ritz. The Managing Partner of my firm spoke highly of the Japanese restaurant tucked into the back of the Ritz, though I haven't tried it myself. Blue Ginger, widely regarded as the top Peranakan (loosely: native Singaporean) restaurant, was unimpressive. To summarize: Singapore is food food food.