How do we measure time? I'm having a little trouble trying to put to words my problem and I apologize in advance for any causation of trouble in trying to interpret it.
We define periodic events as those events that occur over equal intervals of time. But, don't we use periodic events themselves to measure time (like a pendulum or the SI unit definition of transition frequency of Cesium)? Then how is it we know we have equal intervals of time?
Another way to put my problem would be:
We metaphorically describe time in terms of the physical idea of motion, i.e., 'time moves from a to b', but how do we deal with how fast it moves because to know how fast it moves, we must know its rate and to know its rate is like taking the ratio of time with time?
This is all very confusing. I apologize again for any problem in trying to understand.
 A: It's a very interesting question, but hard one with no easy answer. Indeed, definition or measurement of periodic time intervals is a bit cyclic. For example second is defined as 9,192,631,770 periods of ground-state radiation from caesium-133 isotope. But how can you verify period of Caesium-133 atom radiation ? Of course you can calculate radiation frequency then invert it as $T=1/f$, but for doing that you still must be able to count oscillations per second, to find out the frequency. So in the end it's cyclic - you need a higher frequency sampling signal for detecting main signal period/frequency. It goes like this :

In this case by sampling main signal with test signal we conclude that period of main signal is $T = 5t_{sampling}$. We were able to do this only because $f_{test} \gt f_{main}$. Similarly, you can conclude by watching your clock that you can stop breathing for $60~\text{seconds}$, just because you can count your watch ticks (mechanic arrow motions or electric impulses, doesn't matter). This you were able to do only because watch tick period is (a lot) smaller than non-breathing period. If for example, you would be able only to measure hour passing without a finer resolution, then you wouldn't be able to say anything about your breathing periods.
A: Yes, we measure time as multiples of periodic events, such as the ticking of a clock, the rotation of the Earth, or the period of radiation from caesium, and we assume that each of those events has an unvarying duration. In a thought experiment, you might imagine whether it was possible for time to expand or contract, so that, for example, a second today might actually be a different duration compared with a second last week. The key question then might be, how would you know? If the duration of everything changed by the same amount- the ticking of all clocks, all natural processes- you would have no way to tell that time had expanded or contracted. Or, to put it another way, it would make strictly no difference to anything, so the question of whether it occurred would be utterly irrelevant.
A: You need to have a periodic even (a "tick") that you have reason to believe is at least as stable as the accuracy you need to measure.
For example, sunrise/sunset is a natural period that people used since the beginning.  It is easy to count days.  Later, people built other kinds of devices that had periods of (say) 1 second, and discovered that the number of ticks between one "noon" and another is not uniform but varies over the year and averages out over the length of a year and then the cycle repeats.  You could not tell that without a more accurate "tick" to measure it with!
Eventually, it was noticed that atomic clocks were more accurate than the regular period of the Earth's rotation, which changes seasonally due to differences in wind patterns (!), has sudden jumps due to earthquakes, and is gradually slowing down in general.  You would have no way of knowing that if you only had a mechanical Grandfather Clock even if it was carefully tuned so it not only showed exactly the right number of seconds in a calendar year but the swing of the pendulum was seen to be in-phase so it's apparently accurate to a fraction of a second over the span of a year.
At atomic clock is accurate enough that you will get funny results if you don't follow the correct protocol.  I recall an anecdote about someone who was lazy and cabled the signal in instead of physically carrying the clock down stairs to sit right next to the thing being measured.  Someone who worked at a standards lab explained that protocols included sitting in the chair just right with both arms on the armrests and hands in the marked rectangles, before reading the measurement.
In the most accurate modern clocks, the reason to believe it is accurate and unaffected by things we don't account for is the simplicity.  Measuring the behavior of an atom rather than a complex machine, it can be isolated and anything that affects it has to be pretty fundamental.
A: When we think about time, we naturally go to the processes of how we measure it. But time is not the ticking of a clock, or the oscillations fo an atom. Time is the construct we use to differentiate events. (No time based terms in the definition.) Just as space is the construct we use to differentiate objects. We need time to perceive order (1st, 2nd, 3rd, before, after, etc). So need the 3 dimensions of space and the time dimension are the constructs we need to describe a universe in motion. If the universe was static then time would be unnecessary (maybe not even exist) since there would be nothing to order that couldn't be accomplished with the 3 space dimensions.
So all the issues with methods of measuring movement with perfect accuracy as has been discussed are accurate.
A: A professor of mine once defined time as follows:

Time is what a clock measures.

which I assume is an of-quoted anglicization of Einstein's:

Zeit ist das, was man an der Uhr abliest

In other words, you build a clock (we all know what that is) and time is the thing who's change it measures.
Of course you may ask "Okay, what is a clock then?" and for that I will refer you to the other answers here.
A: When you measure the time between events you're doing this:

The falling of each card is an event. If you want to measure the time between the falling of the cards on the left you need a clock. Any clock might as well be the falling of the cards on the right. Because every clock tick is an event. Thus we create an ordering of these events that tells us "how long" in clock events.
The thing here is no one has any idea if the cards really are falling regularly. If they all stop falling we can't tell. All we can tell is what events fall between other events.
But you might think you can watch a video of these cards falling and measure time from that. Sure, but you don't really know if you're looking at a live feed or stop motion animation. All you've done is add another pile of cards to shuffle into the pile.
The events we use to measure time are assumed to be regular. But we don't really know. All we ever know is the ordering of the events.
A: As Marco Ocram describes in the answer above, you will not notice any change in the rate of time inside a closed system.  You can only measure a changing rate of time with relation to some other place where time runs at a different rate.
So, for example if we were to take your house, family and all your clocks and put it all on Jupiter (where time runs slower due to greater gravity), then you would not notice any difference the the rate of time. You would continue to live your life and age and cook a 3 minute egg in exactly the same way.   But if, after a while, you called your cousins back on Earth, then you would notice that your cousins have aged more, and their clocks would have ticked more.  It is only by comparing between two systems that a difference can be measured.
It was this discovery, that time runs at a different rate in different frames of reference, that is a major source of the study of physics today.
A: This question, and the answers thereto, make it seem like there's something unusual about time. There isn't.
Processes with characteristic duration are the set of all repeating or repeatable processes whose periods (or the average of a large number of periods) are expressible as a fixed ratio of any other process in the set. A duration is a number times one of those periods. For consistency's sake, it's good to pick just one process that deviates as little as possible over the fewest possible iterations, in its ratio to the average number of iterations of all other processes in the set. We can use that as a basis value, which is where we get the current SI definition of a second in terms of the transition frequency of Cs.
Processes with characteristic length are the set of all repeating, long-lasting, or repeatable processes whose spatial extents (or the average of a large number of extents) are expressible as a fixed ratio of any other process in the set. A length is a number times one of those spatial extents.
Processes with characteristic mass are the set of all repeating, long-lasting, or repeatable processes whose resistance to acceleration (or the average of a large number of resistances to acceleration) are expressible as a fixed ratio of any other resistance to acceleration in the set. A mass is a number times one of those resistances to acceleration.
So on for any measurable value that adds linearly. There's no ultimate basis value for any measurable quantity, just sets of processes that are found in reliable ratios with one another; among which we choose a basis that's easy to measure very precisely.
You can get at most one step away from this, if you happen to get very good at measuring some related quantity. For instance, experimentalists have gotten really good at making clocks and really good at measuring the speed of light and Planck's constant, so now we use Planck's constant ($h$ has units of $Js$), the speed of light squared (a conversion factor between $kg$ and $J$), and a clock, and declare that a kilogram is the mass of an imaginary process that makes Planck's constant have its standard value, given our definitions of $c$ and $1s$. If people were really good at measuring masses and terrible at making clocks, we'd do it the other way around, and define seconds in terms of Planck's constant, $c$, and the mass of a 1kg chunk of platinum-iridium alloy in a vault in Paris.
Every measurable characteristic can entertain the same meaningless "What if we're all in the Matrix?" style question.
How do we know the tick of the clock is always the same real ultimate divine second, and the universe doesn't just conspire to keep all the ratios the same? How do we know the meter stick is always the same real ultimate divine meter, and the universe doesn't just conspire to keep all the ratios the same? How do we know the kilogram is always the same real ultimate divine kilogram, and the universe doesn't just conspire to keep all the ratios the same? How do we know the electron charge and Boltzmann constant aren't constantly in flux? What if we're all brains in jars in a holographic simulation of the matrix on a computer in a computer in a computer in a universe created last Tuesday where God surreptitiously changes the real ultimate divine length of the meter and the speed of light every time it rains?
A: 
This is all very confusing. I apologize again for any problem in trying to understand.

You're in good company. Time is complicated.

*

*Time in physics shows you an evolution of the concepts of time in physical/mathematical terms.

*There are also current physical interpretations of time, leading to the Problem of Time between general relativity and the quantum regime.

*The philosophy of space and time is very fascinating, and shows you that by far not everyone agrees what time is in the first place.

I cannot really give an answer to your question, simply because it makes no sense to ask how "fast" time flows - whatever time is, "fastness" is a property of something relative to time; you can't compare time to time in any useful manner (as you found out).
As this topic interests you, I can heartily suggest to start with skimming through the linked page on the philosophical aspects of time. Very fascinating, even if it does not give you a direct solution, it gives you a feeling of what the great thinkers have been coming up with in this regard.
A: The new approach about time in physics is that it is a mental construct we use to distinguish events or objects. As someone said before, the information is directly related to the time. The information theory can be applied here to understand time in a completely different way than the classic concept of time as a continuous stream. In this new approach, the Shannon entropy may help make measurements of time. This issue has been the subject of many interesting works in quantum gravitational physics.
atomic clock help link quantum and gravity 
nature of the time
A: The don't exist truly periodic processes in nature. So measuring time by a clock will never be precise.
