How to start process improvement in any company?

It is about to change the culture of the organization. It is a way of living then it must be internalized into each person. As into any other cultural change the organization has to motivate people to do that. It has to be taken as a program/project and the same steps that into any other program/project has to be taken.

Pick one process to work on. Try to find other people interested in helping. Once you have achieved improvements, share this with a wide group within the organization. Hopefully, that will drive interest and involvement for the next processes.

Before any process steps, it would be a good idea to
a) Look at the current process and analyse for gaps and identify why it needs to change.
b) Identify what you think needs to change and get buy in from key stakeholders on this.
c) Set up a meeting to collaborate with people from various (relevant) departments to ensure you get buy in from all and also take into account all that is involved with changing a process
d) Get agreement on what needs to change, the priority of the change and when it needs to happen by and also the cost of making that change (if any).
e) Identify those who have actions to help make that change and agree steps on reaching the goal
f) Proceed with the change and communicate regular updates with all stakeholders to ensure they are kept in the loop and can step in and make key decisions if need be. Doing a RACI may help you. Repeat until change is complete
g) Track the change and measure its performance. Check in with people to see if changes need to be tweaked. Show performance measures to stakeholders and continue to monitor and come up with ideas if things need to be further improved.

Doing as Stéphane suggests is the best way to get started. You need to show people the tangible benefits of process improvement before anyone is likely to invest time and resources in it.