There are, admittedly, other educators who are better qualified to answer this question regarding velocity, speed and distance, but hopefully what follows will help.
Velocity and speed are very closely related, with “speed” referring to how fast an object is moving. That’s pretty basic, as we routinely measure speed in terms of miles per hour when driving. In astronomy, speed is measured in “light years,” the distance light travels in a year, calculated as 186,000 miles per second. The reason for the use of “light years” is because the distances between objects in space are so great that conventional measurements are simply inadequate to the task. For example, the distance between the Sun and Earth is 93 million miles, too close to worry about measurement tools that extend beyond normal numerical calculations. Once one ventures outside of Earth’s solar system, however, distances between objects become too great to employ conventional units like miles, so “light years” are used instead. For instance, the distance between the Milky Way galaxy and its nearest spiral equivalent, Andromeda, has been calculated at 2.3 million light years (ignoring for the purposes of discussion the fact that the two galaxies are on a collision course because they are moving towards each other).
So, the relationship between speed and distance is easily understood: How fast an object is moving and how far it travels, or how many miles or light years (or inches and feet) exists between two objects. The complicated concept is that of velocity, and it is here where the late Sir Isaac Newton comes into play. Because the Earth is round, and because there are environmental factors like ocean currents and wind, velocity has to be calculated utilizing speed as sort of a subset. Velocity, unlike speed, is a “vector,” meaning it has a numerical value that describes an object’s motion in terms of speed and direction factoring in those aforementioned variables of wind, curvature of the Earth, and so on. Vector is subject to constant change because of those forces that influence speed and direction.
At its most basic, we can think in terms of Newton’s First Law of Motion: An object in motion remains in its current state of motion unless acted upon by an opposing force. Gravity, friction, water, wind, and anything else that affects the motion of the object is an opposing force influencing the motion of the object. It may be causing the object to move faster or slower, or to stop altogether, but it is influencing the motion one way or another. The distance between two static objects, as between two houses on the same street, can be measured with a high degree of confidence, as planetary motion and plate tectonics are not having any near-term effects on that distance. (In the very long term, assuming no earthquake, that distance will vary as the planet’s natural motions affect those calculations, just as measurements between the Earth and the Sun will continue to change as the Sun continues its natural evolution through the life of a star, meaning its outermost layers are gradually expanding.)
Returning, however, to the issue of velocity, this is where Newton’s Second Law of Motion becomes relevant. Velocity, unlike speed unaffected by opposing forces, factors in those forces. Those forces can cause an object to accelerate or to slow down. Velocity can measured in terms of the speed at which an object is moving at a precise distance. It is a product of both speed and distance. It can be a constant measurement, or it can measured in terms of changes in the object’s motion resulting from the interplay of forces like wind. Acceleration is an integral component of Newton’s Second Law of Motion, as is “mass,” the amount of matter integral to the object in question. Mass is not directly related to size, as a very small object can contain more mass, and weigh considerably more, than a larger object of less mass or less density. The Second Law of Motion states that “mass” times “acceleration” equals force (F=ma), meaning velocity is determined according to those forces, if any, influencing the object’s movement. What can cause confusion is that fact that both “speed” and “velocity” are determined using the same equation:
Speed equals distance divided by time.
Velocity equals distance divided by time.
The key to understanding the distinctions lies in recognizing the forces that influence the object’s movement.
When I was in a physics class, I hadd a similar issue as you did in terms of understanding the relationships between velocity vs. speed, displacement vs. distance, and time. I'll explain the differences how I eventually understood it, and hopefully that will make sense to you as well.
Firstly, time is time. There is no difference between the time used in physics compared to the time we use in everyday life, besides maybe the fact that in physics all the questions use seconds for time. There is also no "parallel" for time like there is in the velocity/speed, displacement/distance relationships (to be explained later).
Displacement vs. Distance
For displacement, think about it like you are drawing a line between the starting and ending points of whatever object just moved and then finding the length of this line. In other words, it doesn't matter what your pencil/object did in between - you only have to think about where it started and where it ended. Therefore, if you run from one end of a gym to the other end ten return to where you started, your displacement is zero because your starting point (let's call it A) is the same as your ending point (therefore A) and the distance from A to A is 0.
On the flip side, distance takes into account how you got from start to stop. In the example above of you running in the gym, then, the distance would you two times the length of the gym. Rather than only considering the start and stop points, distance takes into account every point that the object has been, making this value path-dependent. Therefore, two objects that undergo the same displacement might have traveled different distances. Again going back to the example, if you actually just stood at point A, your displacement would still be 0, and this time your distance would also be 0 since you didn't move.
Velocity vs. Speed.
The "textbook definition" of the difference between velocity and speed is that velocity is a vector (meaning it has both magnitude and direction), whereas speed is a scalar value (meaning it only has magnitude). However, without a decent understanding of vectors, this definition is kind of confusing.
One way to think about the difference between velocity and speed is to consider their equations. Velocity is displacement divided by time whereas speed is distance divided by time. As noted before, time is just time, so one way of thinking about velocity vs speed is by thinking about the differences between displacement and distance.
Another difference between velocity and speed is that velocity can be negative. Think about an object travelling on a number line from 9 to 5 in 2 sec. The displacement is calculated as the end position minus the start position, and 5 - 9 is -4 units. The distance traveled however, is 4 units. Dividing each of the values by the 2 sec gives velocity as -2 units/sec and speed as 2 units/sec. This shows how velocity can be negative depending on which direction (going back to the textbook definition) is deemed "positive".