I want to get tilt sensor value for a given time t_0, but the tilt sensor data points are discrete. How do I interpolate between the discrete points?
Assuming that the signal is prefiltered and uniformly sampled according to Shannon Nyquist the textbook solution is to upsample it  using some approximation of the ideal (frequency domain) brickwall lowpass filter, equivalent with a (windowed) sin(x)/x function in the time domain.

The problem is similar to showing a smooth (zoomed in) waveform of discretely sampled audio in an audio editor. Perhaps you can find the code you need in eg Audacity?

If the signal contains noise, you might be able to trade noise suppression for signal distortion by knowing the statistics of both and using a Wiener filter.

As this seems to be a control systems project, perhaps you can use a Kalman filter.