1. Enable The HDR function
I've noticed that my photos are a lot better if I first use my smartphone's camera app and then process the resulting image with Instagram. If your smartphone can take HDR photos, like the iPhone, then you can improve your photos even further by enabling this process. High Dynamic Range photos are assembled by using various shots of the same scene taken at different exposures. The resulting 'composite' image will have more saturated colors, and if done well, will have a certain dream-like quality.
2. Keep In Mind Instagram's Square Format
All Instagram photos appear as squares, so when you are taking your shots using your smartphone's camera app, don't forget to keep this in mind. Composition is key to taking good photos.
3. Take Multiple Shots of the Same Scene
If there's one thing that I've learned by using a DSLR daily, it's that you shouldn't be afraid of taking multiple shots. Odds are that at least one of them will be what you want, and deleting the missed shots is not a big deal later. There's nothing more frustrating that realizing later that a shot that you've taken is blurry where you didn't want it to be.
5. Secondary Processing Before Instagram
Once you've got the shot you want, if you want to do some quick photo editing, then you can choose from a number of secondary apps that will allow you to do some light image processing. I like using Luminance since it's easy to use and does everything I need. Once your photo is processed by this secondary app, it's ready to be Instagrammed.
Taking It To the Next Level
If these tips and tricks aren't enough, then you should explore different filters and lenses that are available for your smartphone. There are a number of different ones on the market that will significantly improve the range and quality of the photos you can take with your phone.
(Images: 1. Flickr member Jason Wong licensed for use under Creative Commons, 2. Flickr member Doc Pop licensed for use under Creative Commons 3. Flickr member Greg McMullin licensed for use under Creative Commons)