How can an ordinary person have confidence in the currently accepted laws of physics? With a little ingenuity, someone with DIY skills can replicate classical results such as Boyle's law, the swing of a pendulum, refraction, etc.
Without spending years of study and huge amounts of money, how can an average person convince themselves of the truth of what physicists claim about relativity, quantum mechanics and so on?
I am not positing any kind of conspiracy theory. I am simply asking if there is anything between getting several degrees and taking it all on trust.
EDIT
I see that I might have not been entirely clear. What I am asking is the feasibility of a non-professional who has education up to, say, university entry level but no support from an institution, to carry out home brew experiments that will confirm or refute the results of present-day advanced physics.
Example
@jacob1729 has given such an experiment with regard to polarising filters.
 A: 
Without spending years of study and huge amounts of money, how can an average person convince themselves of the truth of what physicists claim about relativity, quantum mechanics and so on?

I fear that unless the person in question is already extremely technically literate, they wouldn't be able to understand what physicists claim about relativity, quantum mechanics, etc., much less have an avenue to convincing themselves that it's true.
There are certainly experiments that one could perform which in principle demonstrate that certain pre-relativistic models are insufficient. Classical electrodynamics is, in some sense, maximally wrong about the photoelectric effect, and that apparatus is quite simple.  The discreteness of spectral emission and absorption lines can be demonstrated with a $3 diffraction grating and cannot be accounted for by classical models of light and matter.  If the person is willing to accept that Hubble images are not forged, then they can see gravitational lensing for free; if they aren't, they may need to wait for an eclipse, but Eddington's observations are accessible with a telescope and a plane ticket.  Backyard equipment outfitted with a spectroscope can be used to observe the redshift of distant quasars. Superconductivity kits provide a dramatic demonstration of non-intuitive, non-classical physics.  So on and so forth.
However, observing such phenomena is only part of the story.  Presumably in order to be convinced that physicists are on the right track, your skeptic would need to understand classical theories well enough to be sure that their observations are in conflict with them, and modern theories well enough to be sure that they aren't.  Since this is essentially the entire goal of BS/MS programs in physics, I don't see a way for a half-heartedly interested layperson to do it.

I would say that an "ordinary person" probably does not know calculus, but let's allow that up to say 1st year degree level.

This simply isn't sufficient.  How could one really understand why classical electrodynamics fails to account for the photoelectric effect if they can't understand the Maxwell equations? How can they understand that quantum mechanics makes the right predictions if they can't actually use it to predict anything?
At the end of the day, it comes down to the individual.  However, I would suggest that the best a casual layperson can hope for is to apply the same standards to physicists that physicists apply to, say, doctors.  I don't understand my endocrine system, but I am willing to put my health in the hands of a professional who is vastly better educated in medical science than I am, and to accept that some of the explanations I'm given will be on the hand-wavy side.
A: I would be convinced that the laws of physics are true because everyday objects use these laws to make current technology work.
For instance, your quartz wristwatch relies on the piezoelectric effect, the LED lights in your house depend on band theory, your GPS navigation is based on the theory of relativity and so on. So either, all these engineers got very lucky or the laws of physics are correct. This approach doesn't cover all of physics but to be fair, most physicists also cannot claim that they have a deep understanding of all fields of physics.
A: Buy Some Polarising Filters
Take two polarising filters at 90 degrees to each other, they block out all light. Add a third at some intermediate angle (say 45 degrees, but this doesn't need to be precise). Now the combination of three filters, two of which by themselves blocked out all light, allows some light through.
This disagrees with a classical* intuition that an incoming photon is in some definite state that will be blocked by any given polariser. It's simply not possible to say whether a given photon will pass through any of the filters. This is in essence because the photon is in a quantum state which gets probabilistically affected each time it passes through the filters.
I'd also object to the idea that verifying many of the original experiments that led people to quantum mechanics would be expensive. You can probably buy a spectrometer pretty cheaply that would rival high end equipment of the late 1800s which is the majority of what you'd need. It would be interesting if someone (perhaps someone who's set up an undergraduate physics lab?) could estimate how much it would cost to build DIY versions of: a spectrometer to measure the spectrum of hydrogen, the photoelectric effect, Stern-Gerlach experiment, Franck-Hertz experiment.
*For large numbers of photons, the polariser experiment isn't distinguishable from classical wave behaviour. So you might want to either (a) lower the intensity until you are firing one photon at a time through the filters, and record the results on sensitive photographic paper or (b) independantly conclude light is composed of photons from eg the photoelectric effect.
A: Laws of nature vs. law of man
The phrasing of the question is clear by somewhat awkward. Therefore, let me first point out the difference between the nature itself and our concept/understanding of the nature. The laws of nature, including the laws of physics, are not going to change, because we accept or do not accept them, or because we have low confidence in them.
Do physicist simply believe in laws?
Regarding the question themselves: it takes a lot of learning just to understand the laws of nature, as we formulate them nowadays. Convincingly testing them is even more complicated. Moreover, it would take a lot of time. Thus, most people professionally doing physics, accept most of the laws they rely upon without questioning - trusting the authority of those who proved and tested these laws, and wrote books about it. One usually has less confidence in recent scientific articles, where one expect the results to be reproduced by other experimental groups and/or using alternative methods.
Trusting laws beyond physics
This situation of "trusting" is by no means unique to physics. For example, most of the conventional laws (I mean the constitutions, the laws passed by parliaments, etc.) are rarely fully known and understood even by the professional lawyers, let alone average voters, who have neither qualification nor time to study them. Some philosophers conclude from this that not voting is the only moral way to behave, in order to avoid possible unforeseen negative consequences of one's decision. The others argue that the laws should be understandable to everyone, and advocate direct democracy (rather than a representative one), Swiss way.
A: Quantum effects are only observable on very small scales. So you'll have to invest in such equipments to find that particles indeed have a wave nature. I think you could maybe do the electron scattering experiment which some equipment. One upside is that you probably don't have to know the math to verify the wave nature of electrons using this experiment.
Special relativistic effects are observable at very high speeds. One observable effect is the extended life of muons when they fall from the cosmos at high speeds, but that experiment is not readily available obviously. The twin paradox has been tested by super accurate atomic clocks, but that experiment is also not readily available to everyone. Magnetism is a special relativistic phenomenon which every average person has observed.
General relativistic effects are observable in very high gravitational fields. One long standing problem that GR solved was the unexplained orbit of mercury. So maybe you could buy a telescope and see that the orbit of mercury is indeed the one which can be calculated from GR. But you'll have to know the GR math to be able to so any such calculations. Also, maybe gravitational lensing is observable in the sky by home telescopes. You'll have to know the math of GR to verify it by observing the night sky (as you can't detect gravitational waves at your home either).
A: How about the nuclear theory of atoms (that most mass is in the nucleus with electron orbitals around it)? Is that the sort of theory you fear we "just believe"?
This can be demonstrated with a fairly simple experiment that most physics undergraduates would do: The Rutherford Scattering experiment (google for details...) All you need is a collimated detector, rotational slider, $\alpha$-source (e.g., $Am^{241}$) and some gold foil.
Alternatively, you can prove that electrical charge is quantised by replicating Millikan's Oil Drop experiment.
Or the wave-particle duality of light with Young's Double Slit experiment.
Or that the speed of light is invariant with the Michelson-Morley Interferometer.
All these bench-top experiments demonstrate non-classical physics, and can be done by anyone with access to a University physics lab.
A: If not money then mathematical knowledge is a must for such a person. Else such a person will never be able to fulfill their will of verifying the theories.
As you know to verify a physical theory requires the conduction of experiments and experiments generate data. This is where such ordinary man/woman can step in. They can take this data and use them to check whether those are consistent with the theory or not. If such person may question about the reliability of such data then they can verify that with some other independent source of such type of data.
A: To be honest, almost everybody nowadays carries physics micro-lab in their pocket. It's a smartphone !
Just check iPhone 11 Pro specification what it includes :

*

*Face ID sensor

*Barometer

*Three‑axis gyro

*Accelerometer

*Proximity sensor

*Ambient light sensor

*Wireless charging (works with Qi chargers)

*Fast-charge capable

*Voice over LTE (VoLTE)

*Wi‑Fi calling

*Built-in GPS/GNSS

*Digital compass

*iBeacon microlocation

*NFC with reader mode

*Bluetooth 5.0 wireless technology

*12MP camera

*4K video recording at 24 fps, 30 fps, or 60 fps

*Optical image stabilization for video (Wide and Telephoto)

*Triple 12MP Ultra Wide, Wide, and Telephoto cameras

*Ultra Wide: ƒ/2.4 aperture and 120° field of view

*A13 Bionic chip; Third‑generation Neural Engine

*Splash, Water, and Dust Resistant; Rated IP68 under IEC standard 60529

*Super Retina XDR display

*2436‑by‑1125-pixel resolution at 458 ppi; 2,000,000:1 contrast ratio

If this isn't a pragmatic proof about our advanced knowledge about nature laws, then I don't have what to say to you.
A: 
With a little ingenuity, someone with DIY skills can replicate classical results such as Boyle's law, the swing of a pendulum, refraction, etc.

I do not think this is true if there is no recipe to follow, i.e. a do it yourself instruction. Do not forget that it took great minds centuries to move from church dicta to structuring observations into theories.
I believe a person who has enough brains, not only ingenuity, to  find Newton's laws by him/her self would be able to do the same using  quantum mechanical observations, if he/she would stumble on them.
