Unfortunately this thread is marked as level B, although it's clearly at least level I. So I try to answer the question "as simple as possible but not simpler" (Einstein).
The question is, if in any sense we "move" or are "at rest" in the universe. As has been stressed several times in this thread, the first thing you have to do is to find a reference frame, according to which you measure velocities. You must always say in which reference frame you measure velocity, otherwise it doesn't tell you anything.
Now, and that's why I think the "B label" in unjustified, to answer the question we need the general theory of relativity, and that's very hard to explain without math, but I'll try. According to general relativity the geometry of space and time, which together are described as a four-dimensional spacetime geometry, is determined by the energy-momentum content, and the geometry of spacetime is not Euclidean, i.e., it has a curvature, and this curvature describes gravitation. The Einstein field equations determine the geometry of spacetime for a given energy-momentum distribution.
It's also a question concerning cosmology, and in cosmology we have (by assumption!) a preferred frame of reference, because we assume that there's no preferred location or time nor a preferred location in space (cosmological principle). Now math tells us that there is a specific class of spacetimes that fulfill this cosmological principle, the socalled Friedmann-Lemaitre-Robertson-Walker spacetimes (FLRM spacetimes). These are exactly those spacetimes with a maximal symmetry, and there is a preferred frame of reference, where an observer at rest with respect to this reference frame is comoving with the energy-momentum distribution of the universe, which of course must also be homogeneous and isotropic as seen in this reference frame.
But observation tells us that this cannot be true, because we see the stars at the sky, and they are not just one isotropic huge light in the sky but consist of point-light sources. What's homogeneous and isotropic is the large-scale averaged energy-momentum density, and a closer investigation shows that this assumption of a maximally symmetric FLRW spacetime as a model for the large-scale coarse graint few on the cosmos is well justified: There is the cosmic microwave background radiation, which is just the relic soup of electromagnetic radiation from the big bang (which socalled Hubble expansion is by the way also implied by the FLRW-solution of the Einstein equations of GR). In earlier epochs the universe was very hot and dense, and the matter consisted of charged particles: In the very early stages of the elementary constituents of matter, of which we know only a tiny part in terms of the particles in the standard model, the quarks, leptons, photons, weak gauge bosons, and gluons, but that's another story; than a bit later in form of the stable particles known today like protons, neutrons, electrons, etc. but they all were still charged particles. Now electromagnetic radiation is scattered by charged particles and thus this medium of charged particles (plasma) is opaque to radiation as long as it is dense enough. The matter itself is strongly interacting and thus in thermal equilibrium with a definite temperature, but it's cooling due to the Hubble expansion. Now since the photons are scattering also all the time with this dense plasma, it's also in equilibrium forming a socalled "black-body spectrum".
Now at a certain point the universe got cold enough such that the protons and electrons built stable bound states of hydrogen atoms, which are electrically neutral, and from then on the electromagnetic radiation decoupled from the medium, but it's spectrum still stays a black-body spectrum although with ever cooler temperatures the longer the Hubble expansion goes further on.
Indeed, the cosmic background radiation can nowadays be measured very accurately, showing a nearly perfect black-body spectrum with a very isotropic temperature of around 2.73 K, which shows that indeed our visible universe seems to be very isotropic on the large-scale average. On the other hand, the tiny temperature flucutaions of ##\delta T/T \simeq 10^{-5}## provide very important information on the universe and have thus vigorously studied in recent years with a lot of high-precision measurements, most importantly by satellites like COBE, WMAP, and PLANCK, but that's again another story.
Now the last paragraph was a bit simplified, and now I can finally come to your question, whether we are "moving" or are "at rest" in the universe, and this answer makes sense, because we have this preferred reference frame of the FLRW geometry of spacetime, which is defined as the frame, where a resting observer is comoving with the cosmological substrate, and where the cosmic microwave background has a isotropic temperature. Now we can also answer the question, whether we on Earth are moving with respect to this reference frame. Obviously we are, because the Earth is moving around the sun and the sun is moving around the center of the galaxy and whatever other "peculiar" motion all the objects in our direct neighborhood make. The important point, however is, how to measure whether we are moving against the comoving FLRW frame or not, and this is possible again by measuring the temperature of the cosmic microwave background in all directions.
In the comoving frame by definition the temperature is isotropic around each point and the background radiation is described as a black-body spectrum, i.e., it looks precisely like the electromagnetic radiation from a perfectly black body at rest relative to the spectrometer. For an observer/spectrometer moving against the so defined restframe of the black-body radiation, sees this radiation blue or red shifted when he measures its spectrum in a direction moving towards or against the direction of its velocity vector relative to this CMBR restframe. Quantitatively it comes out that in each direction such a moving observer measures again a perfect black-body spectrum in any direction, but he finds a temperature, depending on the direction. The temperature shows a systematic variation with the direction of the spectrometer, which is described by a socalled dipole part of the CMBR temperature variations.
Indeed when the satellite COBE meausured a dipole component in the CMBR temperature variations, which indicated that we move with a speed of around 390 km/s in direction of the Leo constellation. So we can say that we indeed move relative to the comoving reference frame of the large-scale averaged FLRW spacetime.
