If you want to make the most of your time and are there for the diving you don't want to spend it traveling. My recommendation would be to pick a dive resort off the big island and stay there. It's all spectacularly beautiful except for parts of Suva, the capital city on the east coast of Viti Levu, the big island. The international airport is in Nadi on the west coast.
Kadavu is very nice. Great diving. It's the forth largest island south of Viti Levu. I have stayed with Dive Kadavu, Matana (different from Matava) which is the closest to the airport but it would be difficult to say what kind of shape it's in after all this time without tourists. There are a couple of newer more luxury type resorts around but all travel is by boat. No roads really except in the little city there and to some of the villages. It's pretty time consuming to move around.
Taveuni seems to be the most popular. Most of the LDS pick this area. There are probably good recommendations here on accommodations.
Vanua Levu is the second largest island east of the big island. The Cousteau resort is there near the city of SavuSavu. SavuSavu is a cute town where you can find a few other things to do. Two of my best days of diving ever were with the Cousteau operation and they took us to Namena. Namena is world class if you can afford to stay there. As is Wakaya in that same area. Some of the other dives around SavuSavu were hit and miss.
If you decide to stay on the big island, there are multiple little islands around Nadi, the Mamanucas, and you can catch a ferry from Port Denaru. My impression is this better for snorkeling and partying.
The south side of Viti Levu is the Coral Coast. I have stayed at the Shangra La which is a large resort with a golf course and land tours near Sigatoka. The diving there was good. Big sharks, bronze whalers and white tips. No soft coral. Then Beqa is fairly close for the shark feeding. Boats leave from Pacific Harbour near Sigatoka. The Beqa Lagoon resort best dive offering was the shark dive.
Traveling around the big island by car was slow when I was last there. Two lane road not in great shape. Have had wash outs from big storms. Sometimes the island hopper planes wouldn't fly if there was a sign of bad weather.
Like I mentioned best to find a place and settle there for your trip. Especially if you are mainly there to dive. Haven't done a livaboard in Fiji but would love to try it. It's an incredible place so hard to make a bad choice.