I'm new to Kalman filters

I'm new too.

I have no other information (for example, regarding velocity or acceleration).

This is a common situation.

The path of the target is completely random as it modeled by a walking person and it can remain long periods of time just standing still.

 

It seems to me that, although the person can walk freely and apparently randomly, their movements still have a high correlation in a small scale,

In general all objects in the real world moved randomly. But usually dynamic model approximated by some deterministic (quasi-deterministic) model. For example dynamic model of plains approximated by polynomial of degree 1 to describe uniform motion (when this is necessary used polynomial of degree 2 to describe uniformly accelerated motion). Obviously that such models doesn't allow perfectly describe the dynamic model and can be used only for pieces of path without maneuver. To detect maneuver (by speed or heading) used other techniques for example IMM (Interacting Multiple Model). This allow switch current model to more appropriate another. But! Every time you use well defined (not random) dynamic model.

Is a Kalman filter the best approach to smooth the path in my case?

I don't know, I haven't had such experience.

Is this the best approach to solve my problem?

As far you use discrete time and measurement without non-linear transformation this should solve you problem in my opinion.

Also, how would the system be formulated in the case of two dimensions (x,y)?

It would be called the Multivariate Kalman Filter (or may be Two-variate Kalman Filter).

Is there any harm in feeding the same coordinates (x,y) to the filter for long periods of time? (Meaning the person is standing still)

It is not a problem. It just mean that first derivation of $x$ and $y$ is zero.

How does the Kalman filter differ from, for example, the Savitzy-Golay approach to data smoothing?

I don't know, I haven't had such experience.

I'm new to Kalman filters

I'm new too.

I have no other information (for example, regarding velocity or acceleration).

This is a common situation.

The path of the target is completely random as it modeled by a walking person and it can remain long periods of time just standing still.

It seems to me that, although the person can walk freely and apparently randomly, their movements still have a high correlation in a small scale,

In general all objects in the real world moved randomly. But usually dynamic model approximated by some deterministic (quasi-deterministic) model. For example dynamic model of plains approximated by polynomial of degree 1 to describe uniform motion (when this is necessary used polynomial of degree 2 to describe uniformly accelerated motion). Obviously that such models doesn't allow perfectly describe the dynamic model and can be used only for pieces of path without maneuver. To detect maneuver (by speed or heading) used other techniques for example IMM (Interacting Multiple Model). This allow switch current model to more appropriate another. But! Every time you use well defined (not random) dynamic model.

Is a Kalman filter the best approach to smooth the path in my case?

I don't know, I haven't had such experience.

Is this the best approach to solve my problem?

As far you use discrete time and measurement without non-linear transformation this should solve you problem in my opinion.

Also, how would the system be formulated in the case of two dimensions (x,y)?

It would be called the Multivariate Kalman Filter (or may be Two-variate Kalman Filter).

Is there any harm in feeding the same coordinates (x,y) to the filter for long periods of time? (Meaning the person is standing still)

It is not a problem. It just mean that first derivation of $x$ and $y$ is zero.

How does the Kalman filter differ from, for example, the Savitzy-Golay approach to data smoothing?

I don't know, I haven't had such experience.