What is the best STAIN for new concret patio, and best way to apply ?

Contact the people who mixed the concrete and find out what brand and mix they used.... then check at the home improvement store for the best product for the type of concrete... most stains are applied with a "garden sprayer".... you can get them for about $10 and use one for each stain color you plant to use..... use 2 or more for more natural results... and Pinterest.com should have ideas and instructions for you... DIYnetwork.com or HGTV.com too.

We painted ours with concrete paint from the big box store. Used a brush and roller. They will tint it any color you want just like any paint

The stain color is generally added to the concrete before it's poured. Most people paint the concrete.