Vtyjkyuk

I have high interest to professionally know about Euler–Mascheroni constant [I mean more than Wiki page] and to do research on it to develop methods approaching the two unsolved problems about it: Rationality/Irrationality and Algebraic/Transcendental.

I have found no professor in my university (and my city) to help me about this. If I start to study it or to do research on it, I will definitely spend much more on irrelevant subjects than if I had a supervisor.

Would people in here who know about the subject please guide me through the way I have to go?

You can start with a book, for instance: Julian Havil, Gamma: exploring Euler's constant, 2009. It starts at a very elementary level, with a lot of history, but progresses to a lot of results, identities, that are useful to dive into before exploring more technical lands, such as Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, 2013.

Apparently, Stefan Krämer, Die Eulersche Konstante $gamma$ und verwandte Zahlen. Diplomarbeit, 2005, Universität Göttingen is quite cited too, and I could not find an electronic version yet. Yet, he has a webpage on Euler's Constant $gamma=0.577...$ Its Mathematics and History, a work (German version) with over 300 A4-pages and 1250 items of bibliography. And he says:

If you feel the project is interesting you can write to: Email:
skraemer@math.uni-goettingen.de

[EDIT] Some starting points for rational approximations of irrational numbers, continued fractions and Diophantine approximation:

• some classic theorems: Liouville, Thue–Siegel–Roth, Hurwitz, Borel


• Irrationality and transcendance


• 
  Continued fractions, Yann Bugeaud


• The book suggested by @Will Jagy, Neverending Fractions, looks like a must read.

Disclaimer: I am not very knowledgeable in this specific topic, but I do do have experience in self-teaching mathematics.

You are right to be concerned with your question's place on this site. If your question is "please teach me more about $gamma$," then it is far too broad for this site. If your question is "Where might I look to learn more," it's a little better.

I will answer the latter question. Your first step is to read some papers. I would go to Google Scholar or MathSciNet first. Unfortunately, if your school does not have contracts with publishers, a good portion of these results will be behind paywalls. To see free papers, go to arXiv. You can also look at the references section of Wikipedia. Reading these papers will give you a good sense of what problems experts in the field find interesting. You will learn methods, results, and ideas that will be helpful. You will probably also find a lot of things you don't understand. Take it slow; you won't become an expert on the Euler-Mascheroni constant in a day, or even a year. Don't be afraid to take several weeks working through one paper as you start.

Lookup terms you don't know. After a while, you will realize that there are vast fields of mathematics, like number theory or complex analysis, that experts in the Euler-Mascheroni constant have mastered that you know little about. Look for books on those fields of mathematics. Well-worded questions requesting good books here at M.SE are often well received. Read the books. Do the exercises.

If you attend college (and if you are interested in being a mathematician, you should!), find faculty members who do professional research in fields related to the Euler-Mascheroni constant. Seek a position as a research assistant for these faculty members. You many not find experts at your university, but you will find number theorists and others who can help you master the tools you need. Finally, find a problem. It could be one of the problems you have stated, an easier one. Use the skills you have gained to solve this problem. Learn something new that nobody has learned before. Then write papers, and tell us all about it.

If you have direct, specific mathematical questions along this journey, feel free to ask the M.SE community. We love to help!

Good luck.