Sorry about that, ADimitris. The easy part or your questions is combining Raja Ampat and Lembeh. There are daily, or close to daily, flights between Sorong (Raja Ampat) and Manado (Lembeh), so it’s simple to do both on the same trip, given time.
PNG is a major logistical challenge. We went through Australia, then to Port Moresby, and on to Walindi Resort in Kimbe Bay, a good resort where the only two liveaboards in PNG waters are home-ported. However, the dive season changes in late fall from Kimbe Bay to Milne Bay, which is good for muck and pelagics. Oceania’s itineraries move to Milne from late fall until spring, and it would be worth considering joining the liveaboard for a Milne trip; also, there is a good resort (Tufi) there. The waters are vast and truly remote; we saw no other divers (and only one or two other boats, other than dugouts) on a liveaboard trip. The diving is largely pristine. Extreme poverty is the norm. And as to the logistics—we were with three folks from Hawaii whose return home took three extra days.
Fiji is easy to get to—nightly nonstop flights from LAX and SFO to Nadi. Its reputation as the soft coral epicenter is deserve. Gorgeous! Decent macro, but I don’t recall any muck. The only liveaboard in Fijian waters, as far as I know, is the NAI’A. An older boat lovingly converted to a great dive boat. The diving from the boat is quite varied on the longer routes, including mantas, sharks, most all the things. We stayed at Volivoli resort, which we thought wonderful. The draw there is the opportunity to dive the Bligh Waters, an area of current confluence which makes for extremely rich waters, which can be sporty. The Fijian people are lovely and welcoming.
We’re going to Raja Ampat this fall for the first time, so I can’t comment on the diving. But as to your question about economical trips, there are lots of homestays which range widely in cost, amenities and convenience.