Excellent questions. Let me address them in order.
"Does our universe expand in all directions equilaterally?" I'm guessing you meant "equally." Yes, the universe does expand equally in all directions. This is related to the observational principle known as Hubble's Law, which states that the recession velocity of distant galaxies is proportional to their distance from earth. In astrophysics we make this determination using a sort of stepping stone approach. The distance to nearby stars can be determined using the Method of Parallax. The same star, observed six months apart, will appear in a slightly different position in the sky because the Earth has moved with respect to the sun. Using Euclidean geometry and the known radius of the earth's orbit, we can deduce our distance from the star.
But, some stars are far enough away that we can't detect the parallax. That's where a certain class of stars known as Cepheid Variables comes in. A Cepheid Variable, as the name implies, is a "variable star," in that its brightness as viewed from earth - or magnitude, as we astronomers call it - varies with time. For Cepheids, that magnitude has a predictable period. It turns out that the luminosity of a Cepheid variable depends on its period, as shown in the below diagram:
We don't know - a priori - how far away a distant star is. So a dim star that is very close to earth could have a smaller magnitude* than a bright star that is very far away. But in the case of Cepheid Variables, we can measure their distance and their magnitude, and thus infer their luminosity, i.e. the total amount of light emitted by the star (if you know how far away a star is and you know how much light you are receiving at Earth, you can know the luminosity). The period-luminosity relationship was determined by looking at Cepheid variables for which we have parallax measurements. That means that if a Cepheid is far enough away that we can't measure its parallax, we can still know its distance by measuring its period. That lets us look at Cepheids in other galaxies, and know how far away are those galaxies!
Finally, we can look at Cepheids in galaxies that are at moderate distances, and determine their distances based on the periods of the Cepheids. We can also look at the energy spectra of light emitted from those galaxies, and look for spectral signatures. For example, we know that hydrogen electrons emit light at specific frequencies associated with electronic transitions. We know the frequencies by observing hydrogen in a lab. But if the hydrogen atoms are moving at high speeds close to that of light, those emission frequencies are shifted. By observing the new frequencies, we can know the speed of the hydrogen atoms. It turns out that if you look at galaxies that are sufficiently far away, the galaxies are always moving away from Earth. by looking at the nearby galaxies and determining their distance from the Cepheids, you can learn the mathematical relationship between distance and recession velocity. So now you can look at galaxies that are much further away - at distances where you can't resolve individual stars - but you can still know how quickly they are moving away from us based on the spectral signature. See the following diagram for an illustration of this effect:
This effect isn't necessarily true for close by objects. For example, I am not receding from you, and the nearby Andromeda Galaxy is not receding away from us (in fact it is rushing toward us). But at larger distances where the expansion of the universe is more important, all galaxies move away from us.
So to your second question: does this mean the Earth is the center of the universe? No! The interesting thing about this observation is that it is fully explainable by Einstein's Theory of General Relativity. Said theory shows that if you lived in one of those distant, receding galaxies, you would observe all galaxies moving away from you. It would appear that you were the center of the universe. You have probably heard that according to the Theory of General Relativity, space and time are viewed as a single entity, which can be likened to the malleable fabric of the universe. Think of spacetime as a balloon being inflated, and imagine making multiple marks on the balloon with a sharpie. As the balloon is inflated, an observer at each mark will see the other marks moving away from it. Yet no observer is at the "center" of the balloon, and we needn't even imagine the balloon expanding into anything else, because for the purpose of our analogy the surface of the balloon is all that matters (if you ever study differential geometry, this is referred to as a "Riemannian manifold").
tl;dr. The universe is expanding in all directions from the vantage point of the Earth, and that does not mean the Earth is the center of the universe.
*For historical reasons, magnitude is defined so that smaller numbers represent brighter objects. So -4 is brighter than 2 is brighter than 6, etc.