I spent a month driving around Java, Bali, and Lombok last summer, and I did some diving as well so I might be able to provide some pointers.
Regarding diving,
Others have given you good points, the Komodo are the best and Bali + Komodo is totally doable.
The USAT Liberty wreck in Tulamben is incredible and well worth a few dives, a must-do in Bali. There are a few dive shops with accommodation located right on site, otherwise you can look for something in the nearby town of Amed, which has a larger offering and some diving as well. Padang Bai has good diving as well, and is still not too far from Tulamben. I heard good stuff about the tiny Pulau Menjangan island at the very north-west of Bali, but it's a bit more remote.
While not strictly Bali per se, but most people go there from Bali as a day trip, the islands of Nusa Penida and Nusa Lembongan have excellent diving, almost as good as the Komodo. Crystal Bay, Manta Point, etc. are all very good sites, with nearly guaranteed manta rays, and rare Mola mola. If you plan on diving these a lot, it's worth taking base for a few days in Nusa Lembongan -- Nusa Penida is far bigger and has more topside stuff to do but fewer dive operations.
Lombok has the Gilis (Trawangan, Meno, and Air), with nice dives but nothing outstanding. You can, however, find some good deals right now because of the earthquake last summer. I personally really liked the South Gilis islands in Lombok. It's quite a remote and lesser-known area, with only one dive center (DiveZone South Lombok), but the dive sites are lovely and devoid of any crowds. We were often the only ones diving there, and there was a lot of life under the water. If you are looking for more intense diving, you can even go to the remote Belongas Bay, with schools of hammerhead sharks amongst the many things to see.
Regarding non-diving activities,
Bali has a lot to offer, and the North is a good compromise: lots of good stuff to see, but not too many crowds -- by Balinese standards. Temple-wise, and in no particular order, I quite liked Brahmavihara-Arama, Ulun Danu Beratan, Pura Penataran Agung Besakih, Pura Tirta Empul, Pura Gunung Kawi, the whole of Ubud area, and Uluwatu in the very south. The real appeal for me though was simply the outdoors. The scenery surrounding the caldera of Mt. Batur was outstanding, Mt. Agung is a beautiful beast well worth the hike, and the rice terraces are (though too popular) stunning. I would, however, avoid Denpasar and Kuta at all costs. It's polluted, overcrowded, and quite boring. Do yourself a favor and do not base yourself there, you will lose many hours every day in traffic.
Nusa Lembongan doesn't have much to do topside -- mind you, it's still a beautiful place, just small -- but Nusa Penida is amazing. Though it's a pain in the ass to get around, the coastal features are some of the most beautiful combinations of cliffs and beaches I have seen: Kelingking, Banah Cliff Point, Saren Cliff Point (with views of Manta Point, if you are lucky you can even spot mantas underwater), Angel's Billabong, Atuh Beach, etc. The one thing is that it's a pain to travel around, as the one road touring the island is quite poor. If you are into temples, Nusa Penida is supposedly the witchcraft and black-magic side of Bali, so you have quite a few interesting little structures (Goa Giri Putri, and many other spread around).
Lombok has less in the way of temples, but the landscape is still stunning. The South Gilis in particular are a little paradise: it's a lovely archipelago, and the dive center in the area will pick you up from any of the islands. The north-east of Lombok is also a lush paradise, offering many treks and waterfalls. Finally, if you feel a bit adventurous, the Mt. Rinjani trek is world-class, even if you stop at the lake and do not dare to summit the actual volcano.