There's only really one kite spot in Noumea and that's la plage du meridien which is right in front of the meridien hotel on the southern side of Anse Vata. You could walk there from any of the accommodation on the beach front at Anse Vata, or the meridien hotel if you can afford it. There is a groyne and you can either launch there (there's a bit of a sandy beach) and kite on the offshore side. It's a good spot, dead flat water, but often gets crowded with the local hotshots throwing handle passes right near the shore. A bit further north there's a grassy area with a car park and a rocky beach, you can launch here and kite on that side, it's a bit choppy but there's heaps more space. Also, watch out for the coral heads at low tide.
The best place to kite if you're staying in Noumea is Ilot Maitre. You can kite out there from la plage du meridien but I wouldn't do it by myself. Better to get a boat out, but it's pretty expensive (about 3000F, $45 for return trip from Anse Vata) but sometimes the kite school instructors go out there and can take you for cheaper.

I reckon your best to get out of Noumea to some of the other spots in New Caledonia. You can stay at Poe beach, near Bourail, where there is a camp ground and bungalows for rent, fairly cheap, and right on the beach. Big beautiful lagoon but gets quite bad chop. Depends on your level, if you're learning its perfect as can stand up everywhere. If not, it's still a good spot but a bit choppy.

You can also get taken out to some of the islets by boat and camp there, Ilot tenia is great, also Ouanu, next the pass with the left hander. These are great as you can kite the flat water on the offshore side and if anything happens it's shallow enough to stand up (just make sure you don't go out too far). At Ouanu you can also kite the wave.

Also, these islands often get more wind than Noumea, and Noumea gets more wind than Poe.

Have fun

When are you planning to go?
The consistent trade wind season there is Oct-March. I just got an email from some friends there and they've only had 3 kiteable days in the entire month of June...so this time of year is not the best time to go.
If you want lessons, the kite school teaches from Ilot maitre, just google Noumea kite school and you school find their details, the instructor is Antoinne.
Also, there is www.poekiteschool.com, about 2 hours out of the Noumea, they are really great.
But, as I said, they won't be teaching much this time of year.
Also, the spot in Noumea (la plage du meridien) is not a good spot for beginers as its cross off, lots of shallow coral heads to dodge, and gets pretty crowded.