'Catch' is used with moving things: Catch a ball, catch someone as they are leaving. 'Get' usually has the sense of 'procurement'. You get something, then you have it. 'Go get that hat before it blows away'. 'Go get my brother. He can help'. 'Grab' has an implication of doing it fast. Grab her hand before she falls. Grab a bite to eat. 'Take hold of' means to get a firm grip onto something 'Take hold of the handle, and pull'. As far as I know, they are all commonly used, and the usage depends on the context. I guess 'get' is most common in general, but in sports, 'catch' probably is. 'Take hold of' is possibly slightly more formal.

Short words like these tend to have dozens of meanings in English. For example, you can catch a movie, get wasted, and grab a bite. Words like these tend to be less formal, change meaning over time, and have regional and cultural variations. But English does have more universal, stable and formal ways of expressing the same ideas. The police can “catch a crook” or they can “apprehend a suspect” for example. It all depends on what you are trying to say.

We also use 'catch' to refer to contracting a cold - be careful you don't catch a cold - I caught a cold. You can also catch sight of something/someone.