While we can doubt any particular perception, illusions can appear only against the background of the world and our primordial faith in it. While we never coincide with the world or grasp it with absolute certainty, we are also never entirely cut off from it;
The background of the world is the inescapable, shared external reality that all of us who are not solipsists participate in. It is the fundamental construction of experience which is nearly universal among thinkers. It is the undeniable naive realism that arises from the universal nature of the human mind, such as qualia and language, which is inevitably subject to philosophical discourse. One may choose to enter into the process of phenomenological reduction, but one must have something to reduce. This constant assertion of the appearance of reality is this background. The nature of the relationship between the Continental and Anglo-American traditions hinges upon how one copes with this background. If one accepts that one's introspection is the best course of dealing with understanding experience, one is apt to accept rational and introspective epistemic methods as privileged; if one broadens one's perspective to include skepticism of that privilege and rejects that rationalism and introspection are indeed privileged, one must break down those introspections not by thought, but by experiment.
Times, Dichotimzation, and Realities
If you want to understand the background of the world, escape the limits of rejecting testimony, measurement, and experimentation. Start with understanding the relationship between the subjective experience of time versus the objectivity of space-time (even under the aegis of the temporal relativity of relativistic physics); Daniel Dennett has a chapter (6, Time and Experience) in his 2017 edition of Consciousness Explained. It is important to both draw an ontological distinction between the two types of time, and to note the fact that advocates of the view that "time is but an illusion" usually post their thoughts using a CPU with a clock while glancing at their wrist watches that they wear to save themselves time by avoiding glancing up and searching for the clock on the wall before collecting their biweekly paychecks. While it is true that time as experienced is subject to the effects of being constructed by neurons, that hardly makes it random and arbitrary. In fact, time perception and psychophysics are quite robust in their empirical content, and help cut through the hoodoo and voodoo of the relativity of time.
There is much to be learned by introspection, but contrary to Descartes' view that introspection is a very reliable and accurate source of the workings of the mind (it is not), the tradition should be respected accordingly. Unfortunately, some thinkers forget that their thoughts are mere representations of external reality, and not external reality itself. This is best remembered by the phrase "The map is not the territory". You read enough metaphysical speculation and you'll notice self-skepticism seems to be parsimoniously applied to the peculiar brand of generous skepticism which doubts external reality.For a quick sketch of the defense of realism, chapters 7 and 8 in Searle's The Construction of Social Reality.
Time and Experience
The most reasonable position to take on time is one which is ontologically pluralistic, that is to say, that time is both external to the mind (as in clocks and sunrises), and internal to the mind (as in stream of consciousness and flow). If you want to experience life with an awareness of time, wait for water to boil. There's a reason that the saw "a watched pot doesn't boil" has become an idiom. If you want to experience life without any awareness of time, I suggest Tibetan bowl meditation. It sounds silly (and sounds lovely literally), and you can lose a few hours of time. Every night, we dream and usually are quite unaware of the passage of time.
The Construction of Time and Time Dilation
It is true that the Newtonian universe has been cast aside for a more accurate model where time dilates. Time dilation can actually be proven mathematically with trigonometry and vectors at the secondary level of math. Paul Hewitt has a lovely appendix in the 4th edition of his Conceptual Physics like the WP article listed above. GPS wouldn't function without technological accommodations to the relativistic nature of time. But as experienced locally here on earth by people, time is relatively unmerciful and unyielding. It is so real and important, that the progress of society and the fate of the British empire was strongly interlinked to the development of the marine chronometer. Tempo and synchronicity in battle is tantamount to life and death, and is explored in Warfighting by Commandant Krulak.
So, keep in mind Ryle's distinction between knowing-how and knowing-that. On the one end, if you want to understand time better, learn the mathematical and scientific theories of space-time (start with learning the proof for time dilation), and the psychophysics of the mind. Once you understand the science of perception, you'll be able to pair up theory with praxis. Praxis, in this case, means experiencing the variability of time awareness through different activities such as sports, meditation, and the like. But whatever you do, beware of the philosophical mumbo-jumbo that "time is an illusion". That's just a deepity.