When I rendered this image it became really fuzzy and pixelated. I have tried to fix this but it doesn't seem to be working.
Increase the number of samples for rendering.
With a higher the number of samples rendered images will be less noisy and more accurate (read this for more info)
Make sure that you are rendering at 100% resolution.
Resolutions lower than 100% might be pixelated
Set emission shaders to Multiple Importance Sampling.
For more info read this post When should MIS be used and when should it be disabled?
If you are using an image as environment, set it to Multiple Importance Sampling as well.
For a finer control on noise, switch from Path Tracing to Branched Path Tracing
The branched path tracing integrator [...] will split the path for different surface components and will take all lights into account for shading instead of just one. This makes each sample slower, but will reduce noise, especially in scenes dominated by direct or one-bounce lighting.
For additional info please read this post: What is branched path tracing and how is it useful?
Check that the samples settings for the render layers are set correctly.
For more info on this settings read: Cycles render, noisy even when using a high number of samples
Besides the noise, your image shows also a common limitation on cycles default color management settings, in which bright lights desaturate in an unrealistic way. See this post: Render with a wider dynamic range in cycles to produce photorealistic looking images
The whole design principle behind Cycles is to get an image that is complex and realistic, at the expense of either cleanliness or time. It's more for photorealistic levels of detail. For something this simple, I would recommend the Blender Render Engine.
If Cycles is absolutely necessary here, though, there are quite a few steps you can take to make the render time far less painful: http://www.blenderguru.com/articles/4-easy-ways-to-speed-up-cycles/
The image is not displayed at 1:1, hence the pixelation, go to the "View" menu and select "Zoom 1:1".
Strongly saturated and bright light sources (in you case objects with emission shaders applied) often results in noisy renders, so try to lower the emission strength and use less saturated colors.
You can also try to increase the number of samples in your rendered image to get less noise.