Where is the best country in the Middle East to start as a fresh grad in the power generation industry?

QATAR

As a fresh graduate you don not have lot of options to choose from, it is better to start at your own home country.
wishing you the best.

The field of Power Generation can be divided into (Design, Installation & Maintenance) Qatar & UAE may be better at1st &2nd Egypt may be better at Maintenance .
On other hand there is a lot of Power Generation Methods -Diesel Engines , Gas turbines ,Steam turbines & Renewable Energy Methods