Rather than doing an interpolation followed by an FFT, I encourage you to investigate the Lomb-Scargle approach. This will give you a periodogram without interpolation and the additional errors that involves. Google should help. Also, this book isn't bad: "Bayesian Logical Data Analysis for the Physical Sciences: A Comparative Approach with Mathematica Support".

So, first a question: what do you know about the data? For example, is it a sinusoidal signal + exponential + noise? (Just a guess) If you know enough about the signal, you can do a least-squares fit to a model with parameters, e.g. frequency of the sinusoid, exponential coefficient, and then retrieve the coefficients. I would assume that the "noise" other than the sinusoid is Gaussian, but you may know otherwise.

BTW, a Fourier Transform is in fact a least-squares fit to the data by a sum of sinusoids.

The workflow for filtering with a fourier transform is:

1. Apply Fourier to a time series
2. Remove small frequency components
3. Apply InverseFourier to get back a time series

The code below assumes your dataset is called data.

We just want the "y" values for the time series, so we'll be using data[[All,2]]. We don't really need the "x" values - the data points should be and are evenly spaced.

Step One:

fdata = Fourier[data[[All, 2]]];

Step Two:

fdataFiltered = Threshold[fdata, {"Hard", 0.0005}]

or

fdataFiltered = (fdata /. x_ /; Abs[x] < 0.0005 -> 0) 

Step Three:

smoothedData = InverseFourier[fdataFiltered]

Fourier transforms aren't the only way to filter time series. There are a lot of different options for different purposes. WeinerFilter works fine for many cases:

WienerFilter[data[[All, 2]], 3, 0.1]