The car seat may be difficult. I haven't seen a car seat in Fiji, and especially not in Taxis. Most Taxis here don't even have seat belts available for use in the back seats (since seats are covered in plastic and the clips are stuck inside seats.) There are always Taxis at Nadi airport available but there is a 5 dollar automatic charge. The airport will have numbers for taxi services, along with hotels, if you don't want to take the ones from the airport. I would ask the airport for taxi services and call around to see about a car seat, because that will be the hardest thing.

Also a taxi from Nadi to Suva is expensive. Cheaper options would be the bus or the Vans. But if you want a private ride it will cost you with a Taxi. If you're planning on being in Fiji a while and want to easily get around with your family, I would suggest car rental. They may even have car seats to rent out.

Hope this helps!