There are a number of resources on the internet where you can learn stranded/fair isle knitting. One of my favorites is this video from www.knittinghelp.com:

Fair Isle Knitting

I like this video because she is working flat, just as you will on your square. Please watch the video all the through. In the second half she explains how to keep your floats loose to prevent puckering.

Twisting the stitches will help prevent gaps when changing colors. This technique is outlined on this website:

Beyond the Basics of Fair Isle Knitting

For a neat reverse side you can catch long floats every few stitches. How often you catch the floats is up to you, but most knitters will say they like to catch floats that span 3-4 stitches. The two most common ways to catch floats are weaving and twisting. You can use the method of your choice, just be consistent throughout your work and avoid adding tension to the floats as this can result in puckering. These techniques are outlined here:

Tension and Floats

There are many videos on the internet demonstrating these techniques. Here are a few links that demonstrate techniques used to catch floats:

Catching floats on the right side carrying yarn in both right and left hands:

Catching Floats

Catching floats on the right side carrying both colors in the right hand (English):

Catching Floats on the Right Side

Catching floats on the right side carrying both colors in the left hand (Continental):

https://www.youtube.com/watch?v=Kmn01a03LR8

Catching floats on the wrong side carrying yarn in both right and left hands:

https://www.youtube.com/watch?v=GA_Fr1zUgaU

If you are still having trouble with this technique I recommend that you take your project to your local yarn store for some in-person help.