ADXL335 | A library to read the ADXL335 acceleration sensor

Background : I've connected an ADXL335 accelerometer to pins A0,A1,A2 on the Arduino Uno. During the measurement, the Arduino is connected to the laptop via usb, and accelerometer is powered directly by the 3.3V pin on the Arduino. The intention is to read voltage from the accel and plot the corresponding time and frequency spectrum using Python. The overall process is quite straightforward and I'm actually able to achieve this. A snapshot of a signal I recorded is shown below. Arduino code to read/write from the accelerometer and Python code to read from Arduino and construct the plots are provided at the bottom for reference.

The issue I'm facing is with the actual values read by the Arduino (I think).

Problem: There seems to be some noise picked up by the Arduino during the read/write process. To demonstrate this, I ran two separate data acquisitions with the sensor at rest. Snapshots are provided below. I had used a similar system in 7yrs ago in grad school, using National Instruments DAC and LabView. I did experience some noise back then but that was akin to electrical noise ~60 Hz. This one is quite peculiar and I don't know how to deal with it.

Closing: Could someone please help me understand what noise it is, that I'm seeing? Perhaps knowing what this is called will help me research about it in detail and try to tackle it. Any pointers/references would help as well.

Future Scope: I believe that a stable at-rest signal as the first step will greatly help mitigate the offset in Voltage (DC-component in freq-spectrum) even with a simple average. Also, I need to learn how to increase the sampling rate from the uno.

Programs used:

Arduino code to read/write from the accelerometer