I did my open water in Nusa Lembongan. I realise this means I have little experience, but I really liked the dive school so thought I would mention it. It was quiet, and I felt very reassured.

Been diving a couple of times around Bali, with different dive schools.
Nusa Lebongang....very cool dives if you like Manta Rays (manta point) and are looking for the Mola Mola (Blue Corner). Manta point is quite a dangerous current dive on most days, but there can be up to 20 manta's at one time. There's nothing else to see except for manta's on that site! And a lot of the times the sea is too rough to go there!
Blue corner is a nice dive site, with quite some currents...water can be quite chilly! (18-20 degrees)
I went diving in Tulamben aswell....really nice dive site! Nice warm water (26-30 degrees) and a very cool wreck with many things to see! Loved this dive site!
Went diving there with a dive school from tulamben (took a taxi there, and arrived at around 10 in the morning) there's a couple of dive guides there that take u out for a way better rate than all the dive schools in Kuta / Sanur etc!
Furthermore I can really recommend diving in Padang Bai!
Some really nice muck-dives to be found there, and cheap aswell!
I went to manta point with a diving school from Padang Bai aswell....Blue Lagoon divers! Good equipment and a very good and friendly instructor!

I would cut Lombok and the Gilis, and maybe Alor because it sounds hard to get to. Bali sounds good as an introduction (also to the food, the culture etc).
Do not miss Komodo, Bunaken or Lembeh (minimum 5 days each).
To save money, do some snorkelling on Bunaken instead of diving. The reef on the east side is great. You can snorkel from the beach for free. One highlight for me was snorkelling at night, with wetsuit and torch, from the tip of the village pier :-)
UPDATE: Our trip is currently in progress and I thought I'd give a quick report about the diving we have done so far.
We started our trip at Tulamben and stayed/dived with Liberty Dive Resort. Great place. Rooms are very comfortable, food is good, diving is inexpensive. My husband was doing his advanced course with a freelance instructor we liked a lot and I dove along with them, so I can't speak to the in-house guides. But their equipment is good and they are very professional. The staff is incredibly friendly. Diving was great. In addition to the wreck, we really liked the muck dive Seraya and the boat dive Ala Manda (pregnant pygmy seahorse currently in a sea fan). Overall, we liked the diving around Tulamben almost as much as diving we did in the greater Sipadan area, although it doesn't have a site like Baracuda point. But high quality diving with lots of interesting, rare creatures.
We also did a morning of diving in Candi Dasa (Gili Tepekong and Gili Mimpong). We went with Yes Dive because they offered us a really good price ($60 for two dives, rather than the $90ish other places in town were charging). The equipment is really new and works fine, but don't expect much from them overall - our local guide was nice enough, but did zero briefing and seemed pretty disinterested in what was going on with us underwater. He wasn't reckless or anything, but wasn't checking on our air or keeping a close eye on us like the typical guide does (he trusted us, I guess?) I wouldn't say DON'T go with them, but be warned that you need to look after yourself. Gili Mimpong was pretty murky that day but looks like it could be a very nice dive. Tepekong was in better shape and we enjoyed it greatly.
Glad to see you've enjoyed your diving in Bali. If you decide to dive in the Gilis or the rest of Lombok, check out Blue Manta.
In North Sulawesi, I would strongly recommend diving with Bastianos (check out www.bastianos.com). They will offer you great service, but they also offer free transfer from Bunaken to Lembeh or vice-versa, as they have resorts in both locations. I would request diving with Winston in Lembeh and with Donny in Bunaken, if they are both still around. All the other guys are good, but these guys are top-notch!
In Komodo, I've dived with Dive Komodo. They have a new live-aboard boat and it's worth every penny! If you go on the boat, ask if Herry is the one guiding. He was a great guide who catered to divers' requests. We loved Cannibal Rock so much we requested to dive on that site 3 of our 4 dives on a certain day. The food was wonderful, too.
Hope this helps!
Cat

Each dive destination in Indo seems to have it's own character. I did some diving around Manado which was great diving - albeit surphace side was a wee bit boring! Quite resort orientated up there. Komodo is of course great diving with big stuff to see and I thought the town of Labuanbajo was pretty cool - a few nice cafes and a bit of an outback feel to it. Lots of friendly travellers to bump into. Diving on the Gilis? ummm, well I think somebody summed it up in a previous post... diving is something to do for a break from partying there!
Best time I had diving in Indonesia was on Nusa Lembongan off Bali. Diving was superb!!! And it was such a funky, relaxing island that I ended up staying there for much, MUCH longer than anticipated. I met a Canadian marine biologist who was building a dive centre there - Blue Corner Dive (www.bluecornerdive.com) although they were still under construction when I went diving with them, the operation was top form - safety checks, well trained staff, and just a genuinely fun atmosphere to be around. Learned a ton about coral! I'd definately recommend them! They also were building a pretty sweet looking dive training pool.