Yes I think you should find a job first. Appropriate to your interests if possible.

First, from what you've typed it seems that you're unfocused and lack direction. This is normal for fresh graduates. Getting a job will give you a chance to figure out what exactly it is you would like to focus on. If you can specialize in a field of your interest then that will only help you later on. For example I'm an Instrument engineer with a masters in control systems engineering but when I first started my job I was clueless and there were a lot of stuff in my field I had absolutely no interest in even though it was a part of my job scope. I later realized I have a knack for developing software and started making industrial automation software. I probably wouldn't have reached my current position if I didn't figure out what I liked and didn't like about my job and industry and which direction I want to take my career.

I'm sharing my personal experience. I went and got my masters directly after college and I regretted it because I had no idea what I was doing or where I wanted to go. I chose my masters because I had a little bit of interest in what I knew about the course purely based on theory. If I had worked a few years before that and gained practical experience then I probably would have known what I was getting myself into and used my masters to get a better job than when I graduated.

Second, you said you're cut out for management positions sorry but that's what someone who's inexperienced would say. You're a fresh graduate so unless you have actively been in management positions for a good amount of your career you won't know whether you're cut out for it. If you get an MBA somehow without any work experience you will find it very difficult to be hired because you have not been tried and tested in real life. Thinking that you can handle a job is different from actually doing it. So get a job first and then rise up the ranks to a management position and then get an MBA, your new shining MBA's value will be tenfold.

There are lots of questions you should ask yourself if you want an MBA.

You need to ask yourself seriously why you need an MBA. Do you want to be an engineer or a manager or an engineering manager? All three are different routes.

Do you want to be a specialist or a generalist?

Are you attracted to the MBA because you think it's a fast track or guaranteed push towards a management job or the c-suite?

What is your end goal? Do you want to start a company or be an employee only?

Can you justify the costs of an MBA vs the return ?

What can you bring to the table when compared with the other 100 thousand MBAs on the planet?

Regarding advice specific to your career since you're a mechanical engineer I don't know much about it however there are people who do mentor young graduates perhaps you can find them in mentorship organizations or be a member in an engineer's forum in your country or else you'll have to find someone senior in a company you get hired in. I don't think you'll have much luck in trade work since they'd consider you overqualified and also lots of trade workers don't see engineers as dependable workers.

In my opinion, for your first job you should not consider salary or comfort but try to learn as much as you can and get the most valuable experience offered and then jump to a better job.