Hi Julie and welcome to the forum ...
There are a number of layers to the Earth's atmosphere. Basically the "sky" starts right above the ground. If you're not on the ground (or attached to something that is) then you're either in the ground or in the sky. The first part of the atmosphere is called the Troposphere where temperature decreases proportionaly with altitude (for that reason it's also called the gradient layer). This is up to between 8 and 16 (ish) kilometers depending where you are. Next is the Stratosphere. This is about 30 kilometers thick. These two are also called the lower atmosphere. Next is the Mesosphere which is the middle atmosphere and terminates at about 80km above the Earth. Above this is the upper atmosphere with the Thermosphere, Exosphere and Ionosphere.
As to where space begins is a bit of a tenuous ask. The atmosphere thins as you increase in altitude and because of that it's difficult to say exactly where space begins. Even at over 150km above the Earth there are still traces of breathable gases but this area is definitely considered space. NASA, for eg, awards astronaut status to anyone who has flown at or over 80kms above the Earth. So basically Space depends on what you are trying to do. For a better answer you can go
here.
Hope this helped...
Josh