[Vibrant Q&A, Help & Troubleshooting] N00B_IN_N33D:
Here's some tips I found for taking photo's on a phone:

Think of it as a camera

Holding your phone in one hand and trying to press the shutter button will only result in blurry, wonky shots. Holding your phone with two hands quite close to your body will mean your phone is much more stable and as a result, your images will be better.

Stay still

Having your arms tucked close to your sides and standing as still as possible will stop any blur caused by movement from spoiling your shot. Don't move your phone away as soon as you've clicked your shutter button either as shutter lag can mean your camera's still processing the shot so you'll end up with a blurry photo of another part of the scene rather than the person or object you want a photo of.

Use accessories

Many companies are now producing lenses for mobile phones that allow closer optical zooming, special effects, and more. One example of this is the Jelly Lens. There are probably many more, but you can just search around Google to find those.

Zoom with your feet

Use your feet to move you closer to your subject rather than using the digital zoom which can lower the quality of the shot.

Light

You'll be relying on natural light most of the time so get outside, when possible, and set up near windows or patio doors when working inside. If you need to turn a few lights on do watch the colour cast they create. You can tweak the white balance or there are apps available which you can do this with too. There are also exposure compensation options so you can brighten or darken your shot. It's worth checking your settings...

Be careful where you position yourself when shooting outdoors as shooting in the direction of the sun can result in flare, although some people like the effect and it can actually work quite well with some shots. As well as having too much light in your shot too little can result in grainy shots where not much detail can be seen.

Check your settings

Go into your camera's settings and make sure the highest resolution your camera phone can shoot at is selected as the higher the resolution the better the quality of the shot is. If you can adjust the image quality make sure this is set on the highest setting available too. You may also find there are various shooting modes on offer as well as pre-sets such as night and portrait mode all designed to help you take the best shot possible.

The lens

It's quite easy to take a shot where a finger or thumb is covering the lens so watch where you position your hands. As phones are put in pockets and carried around in bags they get dusty easily and due to us handling them a lot they get covered in greasy fingerprints quite quickly too so make sure you keep your lens clean and check it before you take any shots.

Try different angles

Working with a phone generally gives you more flexibility than a camera that's on a tripod so make the most of this freedom by varying your shooting angle and trying new shots. People don't always know you're taking a picture when you're holding your phone either, making them good for quick street candids. This doesn't mean you can take photos in places that say you can't though, you still have to obey by the rules.

Don't forget about post production

Using a phone to take photos with isn't an excuse for not correcting wonky horizons or adjusting the brightness levels in a shot. You can edit your photos with apps on your phone but the screen is a little small so don't overlook transferring them to your computer so you can edit on a larger screen.

Experiment with apps

There's no excuse for not experimenting with your phone photography as the many apps that are available mean you can colour pop photos, add frames, create film effects... the list goes on. There's also apps out there that offer you shooting advice in the palm of your hand so if you want to brush up on nature photography while out and about, for example, you can. Some apps you have to pay for but there's also many free apps available which are just as good. Some good apps are FxCamera, Paper Camera, Pixlr-o-matic, etc.

(Don't forget to get out, as nice as your house may be, the best shots are awaiting you outside...)