First things first: like the term "christian" you do not need to capitalize "philosophy" unless it is at the beginning of a sentence.
How to start Philosophy and find the branches that are related to my questions?
Philosophy translates from the Greek through the Latin to love of wisdom. This simple etymological fact has remained unchanged for nearly 3,000 years. A good starting place is to have a clear understanding of what philosophy (read: love of wisdom) means.
In the context of the initial utterance, philosophy distinguished itself from sophistry, or, the professional dispensation of advice or wise counsel (a contemporary sophist in the non-pejorative sense would be a lawyer or doctor, or, pejoratively, a con artist, a palm reader, a snake oil salesperson). The value of this counsel was often relative to the price paid for it or to the status of the person paying for such counsel.
By contrast to sophistry, love of wisdom, however, is no mere amateur pursuit. In the context of initial utterance, love was akin to virtue, respect or reverence, and, had nothing to do with modern notions of romantic passion except that you could say philosophy is the pursuit of wisdom. In the sense of the initial Greek meaning, that which philo- is respect for, reverence or virtue of, however, is the point: wisdom.
Some consider wisdom the intelligent application of knowledge, but ask yourself, intelligent according to whom? Donald Trump? Alex Trebek? Noam Chomsky? Mamma June? No, wisdom requires knowledge, not opinion, sentiment, or personal points of view (if you say otherwise, you need only show how). Wisdom, all wisdom, simply obtains knowledge. In this sense, philosophy is the respect for obtaining knowledge.
Is there such a thing as fundamentals of [p]hilosophy, the same way that we have in [p]hysics and [m]athematics?
The purview of philosophy (read: respect for obtaining knowledge) is logic, rhetoric and reason. Fundamental to this purview are the excluded middle, non-contradiction, identity - often referred to a the three "laws" of thought (tho do not be misled by this grandiose title for axiomatic primitives).
The domain for the virtue of obtaining knowledge is epistemology and ontology. Epistemology is the study of epitēmē - the study of knowledge. Many would like to imagine that knowledge is "justified, true belief" however this is not the case. Knowledge is empirical verification of what is (the case, states of affairs, the world). Ontology is the study of ontos - the study of existence (what is, the case, states of affairs, the world).
Some have suggested that where knowledge is not to be obtained that reverence for obtaining knowledge simply rejects false arguments and militates against willful ignorance. If the false argument is to be rejected, how then do we rationally assess a true argument? What then is truth? Again: do not be misled by grandiose titles and speculations. Truth is mundane - also needing no capitalization except at the beginning of a sentence. Truth is merely a condition of propositions (sentences, statements, utterance.) When utterance corresponds to (matches, fits) what is, the utterance is true. What is is that which is empirically verified and knowledge is the empirical verification of what is. Truth is merely a condition of statements about what is.
I am interested in topics that is related to the questions that if the reality that we perceive is the only thing that exists, for example, could we be in a simulation? Are we just brains in a glass?
A great place to start is with John R. Searle's refutation of "the bad argument" in his article, "Perceptual Intentionality."
The other topics I like to know is that if knowledge is fundamentally limited. For example, any theory of science has to start either by an axiom or an empirical observation. The axioms of geometry for example cannot be proven. Do we have the capacity to understand whatever there is about Universe? or Is there a limit to understand the physical laws of Universe or Mathematics itself?
Inevitably you will confront someone playing the "absolute" knowledge card. Yes, knowledge is limited, imperfect, incomplete and certainty only a mood. Despite this imperfect knowledge we build bridges and skyscrapers, replace human hearts and send men to the moon and back.
Yes, science presumes a naive or direct realism. Is there any sound reason not to?
Proof is simply a matter of convincing a sympathetic audience.
Get to work on logic, reason and rhetoric and you may find that the way you pose these questions changes, your inquiries refined and your aims ponderable with answers obtainable by heuristic virtue.
In closing, note that the history of philosophy is not philosophy. Except as they are descriptive, metaphysics are not philosophy: they are poetry. This is not to say they are meaningless, just that the whole of metaphysics is little more than pernicious nonsense frequented with tedious regularity and not so much as one single iota of relevance to advancing knowledge claims or confirming hypotheses. If you must read the works of philosophers, read some Hume, Ayer, Russell, Whitehead, Wittgenstein, Popper, Austin, or Searle - and have fun!
People say again and again that philosophy doesn't really progress, that we are still occupied with the same philosophical problems as were the Greeks. But the people who say this don't understand why it has to be so. It is because our language has remained the same and keeps seducing us into asking the same questions. As long as there continues to be a verb 'to be' that looks as if it functions in the same way as 'to eat' and 'to drink', as long as we have the adjectives 'identical', 'true', 'false', 'possible', as long as we continue to talk of a river of time, of an expanse of space, etc. etc., people will keep stumbling over the same puzzling difficulties and find themselves staring at something which no explanation seems capable of clearing up. And what's more, this satisfies a longing for transcendence, because in so far as people think they see the 'limits of human understanding', they believe of course that they can see beyond these.
Ludwig Wittgenstein
"Culture and Value"
pg. 15e c.1931
