What kind of patent should be applied for the improvement of machinery and equipment?

It is more likely to apply for a patent for utility model as you said.

Application for patent for utility model

The plan:

Application-Acceptance, Payment-Preliminary Examination-Grant of Patent Right-Issue

Patent license

To apply for a patent for utility model, the application documents shall include:

The request for the patent for utility model, the abstract, the appended drawings of the abstract (if applicable), the specification, the claim and the appended drawings of the specification are in duplicate.