Need 3-axis wireless gyroscope/accelerometer.
You chose a pure accelerometer. It is best for measuring linear acceleration (including gravity), but for rotation around the vertical axis you should rather look for a gyroscope (measures rotation) or a magnetometer (measures orientation to north). There are many boards available (you can find some here https://www.sparkfun.com/pages/accel_gyro_guide), but you have to consider your required accuracy.

The second requirement is more difficult: According to Newton's law you can not decide if an object is moving with constant velocity or not, because no external force is applied that could be measured. So you have to consider side effects. For example:


If your object is moving, there must have been a previous acceleration that you can measure with an accelerometer. You can integrate over the acceleration to get the current speed, but that is quite noisy and error prone.
Most movements come with vibrations that can be measured with an accelerometer.
You can measure the current position and therefore your speed with a GPS module. This is only useful outdoors and for higher speeds.


There are other options (like time-of-flight measurements of radio waves), but I would try one of these first.

The third requirement is a radio connection. I would start with a Raspberry Pi and a WiFi stick, because it is quite easy to set up and has a reasonable range and data rate and you can connect your sensor directly to the Raspberry Pi.

EDIT: According to the commets, you want to measure movements of your legs and feet. This narrows down the possibilities:


With an accelerometer it is quite easy to determine if someone is moving or standing still, because the foot is constantly accelerating and decelerating.
Getting the current speed is quite difficult - actually when walking, a foot has v=0 over ground half of the time.
But you can analyze the acceleration pattern and try to count the number of steps.
Since you want to attach it to your leg, an Arduino might be more reasonable because of the reduced power consumption, but it will make the wireless connection more difficult.
Buy an IMU and a GPS module and try how far you can get.