What is the best method to fit a curve for linear type of data. It may be noted that there is only one feature and one target. Number of entries are around 20.
It seems like a straightforward linear regression would do the trick for you. You can find tutorials for this in virtually all commonly used programming languages, or even in Microsoft Excel (I believe it's called a "trendline" in Excel). If you feel that this linear fitting is not appropriate for your data, you could look into nonlinear regression.
I believe you would be better suited asking these questions using a search engine, as opposed to on this platform. This isn't the best platform to teach, say, machine learning, and there is an enormous amount of information online about ML, data analysis, curve fitting (which as Professor Dokov showed, are really the same thing). I don't know what parameters a, b, and c are, these sound specific to your problem and therefore we can't really help beyond pointing you to some resources.
To your original question, the "best" method of curve fitting depends on your application, there is not usually a one best method for anything. I think starting with a linear regression and then doing a nonlinear regression would be the place to start. So, you would want to search "linear regression in X" or "nonlinear regression in X", where X is the programming language or other software you are using.
If the relationship between your x and y is causal, please try multidirectional least squares regression. The only software that has this implemented is this: