I went last May and it was fantastic
Used a Nikon D60 and 55-200mm short zoom lens (as that was all I had at the time) but still got some good shots as the birds are really tame and come very close. No doubt I'll be going back again this year with my improved kit, as I've bought my Bigma and D300 since then, so I can't wait!
The best area (and the only one where you'll see Puffins in huge numbers) as The Wick, and the birds are so close that they're literally getting under your feet at times! There are Chough on the island as well, and I was also lucky enough to spot a little owl on a wall near the old farmhouse. The short eared owl are best seen in the marshy area in the centre of the island, although sadly I didn't see any, probably because I was only there for the day and they tend to be more active during the early evening once I'd left
I went on the 2nd or 3rd weekend of May, and there was lots of nest building going on, with Puffins heading in and out of burrows with beakfuls of twigs. I think if you went a bit later, June sort of time, you'd have a better chance of getting those classic "beak full of sandeels" shots!
The puffins are incredibly easy to photograph, but to get something different, just sit and watch at the Wick for a while and work out the routes they take to their burrows, and the points where the land on the cliffs, as you can get some fab landing shots that way with their orange feet sticking out and looking comical
You can get these either by tracking flying puffins, or manually focussing on a cliff edge and then pointing the lens at an area just above it where the puffin is likely to fly into, and shoot when it does! It's a bit hit and miss but you can get some nice shots this way too.
As for camping...Newgale is pretty good and has a pub right next to the campsite, but it does get a bit full of surfers. I think there's a nice site at Caerfai as well (just South of St Davids on the coast).
Another great place for wildlife is Bosherston Lily Ponds, where you've got a good chance of seeing otters, the lilies are spectacular in early summer, and theres a fab beach nearby too, plus loads of chough in the area. If you wander further round the coast you reach St Govans chapel, which is built in a cleft in the cliffs and well worth a visit, and even further is Stack Rocks, where you can see huge guillemot and razorbill colonies.
For red kites, The feeding centre at Llandeusant is worth a visit and is vaguely en route to Pembs
Feeding station for Red Kites bird watching in Wales
The WWT centre in Llanelli is also fantastic for bird photography
Hope that helps!