Firework photography is something I really enjoy having a go at whether it's the local Scouts evening or something larger such as Leeds Castle. I've tried hand-holding, on a tripod, with a DSLR and a compact camera, whatever your equipment it's well worth having a go although the DSLR's do make life a tad easier in their myriad of setting choices.
Handheld I've used Aperture Priority mode primarily at F5.6 with ISO set to something pretty high, like ISO3200 and just let the shutter speed be set by the camera. Hit and miss as these subjects tend to be but if there's no option to use a tripod it's as good a starting point as any.
On a tripod, for something such as the Leeds Castle displays I use Manual Mode at F16, ISO200 and a shutter speed around 4secs to capture the bursts - this seems to work pretty well and you can always alter the shutter speed to suit, the lens can be covered with some black card in between bursts if needs be but you need to ensure the image doesn't burn out. The aperture of F16 ensures that any lights shown during the display come out at star bursts - so quite a nice touch. Use a remote as well to fire the camera to ensure you eliminate any possibility of camera shake.
1/10 sec at F5.6 ISO3200
Hand-held at The Needles Park IOW
1/6 sec at F5.6 ISO3200
Hand-held at The Needles Park IOW
1/13 sec at F5.6 ISO3200
Hand-held at The Needles Park IOW
4secs at F16 ISO200
Tripod at Leeds Castle
4secs at F16 ISO200
Tripod at Leeds Castle
4secs at F16 ISO200
Tripod at Leeds Castle
So there you have a small collection of some of the firework images I've taken over the years - always a fun subject and don't forget to watch some of the show outside of peering through the viewfinder!