Technically, you can escape Kerbin's gravity at any height, as long as you're going fast enough.
To orbit the planet at any given altitude, you only need to be travelling at the corresponding horizontal speed, as shown in the table below. The only reason that you can't orbit the planet consistently at low altitudes is that it's difficult to maintain your orbital velocity in the presence of atmospheric drag.
For example, to enter an orbit at 100km above the surface, you need a horizontal speed of 2245.8 meters per second.
Kerbin's atmosphere tapers off at approximately 68-70 km above the surface. Any orbit below this altitude will require corrections due to atmospheric drag, or else your ship will begin losing orbital speed and eventually deorbit. You can also do this intentionally if you want to deorbit your spacecraft - reduce your speed enough to drop below the minimum horizontal speed given in the table, and your spaceship will begin to descend.
I've snipped portions of this table to save space. The full thing can be found on the Kerbal Space Program Wiki:
Altitude (m) Horizontal Speed (m/s) Orbital Period (min)
35000 2357.9 28.20
40000 2348.7 28.54
45000 2339.6 28.87
50000 2330.6 29.21
55000 2321.7 29.54
60000 2312.8 29.88
65000 2304.1 30.22
70000 2295.5 30.56
75000 2287.0 30.91
80000 2278.6 31.25
85000 2270.3 31.60
90000 2262.0 31.94
95000 2253.9 32.29
100000 2245.8 32.64
110000 2229.9 33.34
120000 2214.4 34.05
130000 2199.2 34.76
140000 2184.3 35.48
150000 2169.6 36.20
160000 2155.3 36.93
170000 2141.3 37.66
180000 2127.5 38.39
190000 2114.0 39.13
200000 2100.7 39.88
300000 1980.6 47.59
400000 1879.0 55.73
500000 1791.5 64.30
600000 1715.3 73.26
700000 1648.0 82.61
800000 1588.0 92.32
900000 1534.2 102.39
1000000 1485.5 112.79
2,868,378 1008.9 6 hours
8,140,000 635.4 24 hours
The SAS module is designed to keep your craft on its current course by means of altering fins and micro-rockets.
You should turn the SAS off while you maneuver your craft onto the desired heading, then turn it back on to have it automatically stay on that heading.
If you find the SAS is not staying on course, you may need to make your craft more maneuverable with the addition of extra fins or micro-rockets.
Best Answer
You're having trouble with staging, not action groups.
Along the right side of the screen during rocket construction, or the left side during flight, are your stages. Each orange bar is the container for one stage. When you press the staging button, everything in the next stage activates. You have your engines and your parachutes in the same stage, so it's doing what you're telling it to: activate them both at the same time.
If you hover over the staging, you'll notice [+] and [-] icons appear. Press the plus icon to add a new, empty stage. Move your parachutes to this new stage. Then make sure the parachute stage is above your engines stage, if necessary, by clicking on the orange bar and dragging the whole parachute stage to the stop of the stack.
Staging is a key construction tool and as important as how you put the rocket pieces together. Once you know how to use stages, you'll find that getting staging just right is as much a part of the fun of the game as building the rocket itself.