Welch's method divides the data string into short segments, applies a window to each prior to the Fourier transform, averages the transforms, and takes the squared magnitude to produce a power spectrum. It reduces the variance of the DFT spectral estimate (just as Bartlett's method does) and at the same time smooths the estimate in the way windowing always does, at the expense of degraded resolution (also in the way windowing always does). Another advantage over the regular Bartlett periodogram is that the power spectrum is always positive. Most spectrum analyzers use this method.
Answering your other questions requires a deeper understanding of spectral analysis. There are a dozen well-established methods of spectral analysis. DFT-based ones are the simplest and most common but not necessarily best. Model-based methods (AR, MA, ARMA, etc.) are better if you have a model and the time to develop the specific analysis. Non-linear methods (Capon method or minimum variance distortionless response, maximum likelihood, MUSIC, etc.) have far better resolution but are complicated and require a high state of knowledge to use wisely. Wavelets and so-called fractional Fourier transform methods are good for signals with components of varying finite durations (speech, say). Maximum entropy and Bayesian methods are at the top in terms of accuracy, resolution, freedom from artifacts, and also in terms of difficulty and required knowledge. They are seldom used.
The wording of your questions suggests that you are not knowledgeable in these methods. Therefore, go ahead and use a simple FFT-based method. You have plenty of company--probably 90+% of engineers know of and use nothing else.
