There are only a few major changes in philosophy required of learning fundamental physics:
You need to accept positivism. This is the idea that if you have a philosophical question, and whatever the answer, it makes no difference to anything testable by the senses, then the question is meaningless.
Positivism is the only useful thing in philosophy in all the 3,000 years that philosophy has been going on, and it didn't really come from philosophy, it was formulated by Ernst Mach to explain to philosophers what it was that they were getting wrong.
It was accepted in philosophy for a while, but it's bullshit eliminating properties and it's association with communism and fascism made it politically undesirable, and it was purged from philosophy in the 1970s/1980s.
Entropic arrow of time
People tend to think of time as something that is pushed forward psychologically, while physics treats time as a dimension of space, so that it "exists all at once" (although the preceding statement is positivistically meaningless). These two conceptions are at tension, so you get constant babble about physics not taking into account that "time is essentially different philosophically" from space, or that "it takes energy to constantly push time forward", etc, etc.
This is resolved when you switch to positivist thinking, since there is no positivist formulation of the statement that time goes forward, other than the statement that the mental computation has a direction of memory which is determined by the direction of entropy increase.
This philosophical leap was made in the 18th-19th century, and was cemented by relativity.
The philosophical change here is enormous--- it requires you to stop thinking of the world as having a definite state, but to think of the world as emerging from an interaction between observations and microscopic subsystems. The only way to navigate this is positivism, but there is more here, including the renounciation of the idea of a separate variables that describe systems and observers in the case that tehy are entangled together.
This is philosophically challenging, and this revolution in philosophy is still not incorporated into any of the non-mathematical blah-blah philosophy, which makes all that stuff useless and primitive.
This is the philosophical position of S-matrix theory and string theory, that the space and time we see is only reconstructed from variables that only make sense at the boundary. This is also philosophically difficult, and the philosophical rejection of S-matrix physics led many prominent physicists, including Richard Feynman, to reject string theory.
The basic idea is that the only observable things are associated with asymptotic quantities, ones that are defined on a screen far away. So that if you are making a measurement of particles colliding to make a black hole, it doesn't make sense to ask "When exactly did the black hole form?" "Which particle crossed the horizon first?", "what was the first particle to be emitted in the Hawking radiation, rather than the scattering?"
All these detailed questions are meaningless. The only real question is "what is the S-matrix for shooting the particles in and getting new particles out".
This requires that one reformulate all of physics from the bottom up in terms of asymptotic quantities. For the S-matrix, this was done in the 1960s, only to be completely rejected in the 1970s, and painfully gradually revived throughout the 1980s. For curved AdS spaces, the analogous quantity is the boundary conformal field theory, and this was discovered in the late 1990s.
The philosophical position is very radical, since it involves building space time from more primitive ideas at the boundary. This is the most philosophically challenging idea in physics beyond quantum mechanics.
I don't think there are any other major leaps. The only real thing you need is to not read philosophy and to accept positivism.
The Induction Problem
These philosophical shifts are all fine, but I think that the major one you are worried about it how one can acquire certainty of nature, when one has to make assumptions to interpret experimental data. this is called the "problem of induction" in philosophy, and it is essentially completely solved by the existence of computers. This stuff is usually attributed to Solomonoff, and is called "Solomonoff induction", but it's philosophy, and so it was floating in the air.
Once you have computers, then you can define the complexity in terms of computer program length, and you have a definition for Occam's razor--- the shortest computer program that explains all observations is the right hypothesis.
Then you can make a precise notion of scientific induction--- you match theories to data with computational complexity as the guide for selecting theories. In real life, our brains will know when the complexity is large or small, and we would just call this common sense, that the simple explanation is correct.
So this resolves this philosophical question for good. This is also not accepted in the political world of academic philosophy, so another reason to ignore everything they have to say.