The problem is the "small" part...
The bigger the sensor of the camera, the more "information" it can absorb, thus the better the picture.
Think of it as a painting Canvas. You can paint a picture in an A4 paper and then stretch it to fit a wall, the result will be "pixelated" with a digital camera.
But if you "paint" on a bigger frame (as the full frame digital cameras and medium format digital cameras) the result is gonna be better.
This kinda depends on the use of the photo. Going to be only viewed on screen or maybe printed and if yes, in what size?
About the "grain":
In the old days of film, there were different "grades" of film sensitivity, according to how fast they were reacting with light. This carried on to digital with various "ISO" settings.
The darker it gets, the more the shutter needs to remain "open" in order for the "Sensor" to collect the "information" (or light). This might result in Ok, but "shaken" or "blurred" pictures if you are not using a tripod. If you are using one and nothing moves, (wont happen
), you might end up with a crisp but dark image.
With a higher sensitivity, the shutter will need to remain open for less, but higher sensitivity images are ase you said usually grainy. You can maybe "blur" them a bit later on photoshop.
Higher sensitivity in digital, means more electric charge on the sensor, which also means higher battery consumption.
Then is the other element, the lens.
The more light a lens allowes in (that would be the f 2.8 written on the camera lens) the better. For example a f 1.4 lens will allow more light in then a f 2.8 lens. It will also cost 1000 euro alone.
There are now some "new" cameras hitting the market. Called "Mirrorless". Since they no longer use a mirror for the view finder to display the image, they manage a more compact size and they still have interchangable lenses.
Anyway for night photography:
1. A lens at least f 2.8. No zoom. (You can just crop the item later)
2. A tripod might be handy, but it has to be 3 x the weight of the camera to be effective.
3. Release the shutter using a timer. Not the shutter button. This reduces shaking when the shutter remains open for longer.
4. The longest a shutter should remain open, is 1/lens length. For example, with a 50mm lens, the maximum allowed length of exposure on a hand held camera should be 1/50sec. For slower speeds then that you need a tripod.
5. Anything more then ISO400 it's probably a bad idea, with 400 being bad allready. 200 is Ok-ish. 100 is better. 80 with flash is probably best.
6. and probably the best idea. A flash unit!