"Take a rest" or "have some rest"?

In my (UK) idiolect "have a rest" and "get some rest" are idiomatic, not the two phrases you give.

I find a slight difference in meaning. To me, "have a rest" is neutral: I am going to rest, with no implication about how much I need it.

"Get some rest" implies that I have a particular need for rest: either I have been short of rest recently, or some major activity is expected so that I need to be well-rested before it, or there is very little time available for resting, so I will avail myself of it.

Common American English would use "take a break" or "get some rest", using "rest" as a noun. It is almost never used as "take a rest" by native English speakers in the USA. However, this phrase is very commonly taught to and used by EFL/ESL learners in Asia.

I would never use take a rest. Rest itself means physically taking time relaxing, so I would rather use rest just as a verb. And as a noun I would say get some rest.

I would say "take a break" for a brief interval, and "get some rest" for a longer period.