Can we find actual rest mass of things on Earth Earth moves around the Sun and the Sun moves around the galaxy and the galaxy moves with unknown speed and direction. We have speed so the mass of us all altered.
Can we know the real rest mass? If so, can we deduce our speed in the universe?
 A: The rest mass of an object, by definition is the total energy of an object as measured by an inertial observer who is at rest relative to the object. If the object is not moving uniformly, then you can measure its rest mass from a momentarily co-moving freefall frame. This rest mass is also the constant in Newton's law, i.e. the inertial mass as well. 
So you need to be at rest relative to the object, and you need to be confident that so-called tidal effects are not significant over the object's spatial and temporal extent (i.e. that special relativity applies to the object and the comoving laboratory you fit it into, or, from a Newtonian perspective, that the gravitational field over this laboratory can be treated as uniform).
Once you fulfill these mild conditions, then you can measure the object's rest mass, most easily by giving it a whack of known impulse and measuring the change in velocity that follows. So, yes, the mass we measure in our laboratories of things here on Earth is going to be extremely near to the "true" rest mass, with only fantastically small tidal effects tainting this measurement. But, as in Acid Jazz'z Answer, we don't have theories that give us rest masses of objects.
A: We do have a rough idea of the relative speed and direction of our galaxy, with respect to the other galaxies around us, the so called local group.
In general relativity, which is our best theory of the universe to date, there is no such thing as absolute speed, as it depends on which frame of reference you use to measure things in.
Our Earth and the objects on it are composed of atoms, as I  am sure you know, and these atoms in turn are composed of electrons, protons and neutrons. 
The "real" rest mass of these objects is determined by experiment, as we do not currently have a theory that predicts their mass. 
To keep this answer from going on too long, (and to avoid trying to explain things that other people can do better than I can),  I am skipping over areas such as renormalization  (with respect to elementary particles) and dark matter, (with respect to our galaxy).
A: Earth moves around the Sun and the Sun moves around the galaxy and the galaxy moves with unknown speed and direction. We have speed so the mass of us all altered.
The relativistic mass is altered, but this is a somewhat archaic term these days, and is said to be a measure of energy. Nowadays when we say mass without qualification, we tend to mean rest mass. Like Rc and Jazz said, this doesn't change with speed. Instead it changes with gravitational potential, see mass in general relativity and the mass deficit. Unfortunately rest mass is also called invariant mass, which is rather confusing.    
Can we know the real rest mass?
Yes, because the mass of a body is a measure of its energy content, and energy is conserved. But we have no accepted theories for that at present. For example the first free parameter of the Standard Model is electron mass.  
If so, can we deduce our speed in the universe?
We don't need to deduce it. We can measure it. From the CMB dipole anistropy. We're moving at 627±22 km/s relative to the reference frame of the CMB. Or relative to the universe as a whole. 
A: When the object is an elementary particle or a charged ion we can use electromagnetic interactions to measure its rest mass, given the charge in an e/m experiment. One can get the charge with Milikan's oil drop experiment. Here is a setup for the lab.
A: The answer is an emphatic yes!  We can find the actual rest mass of things on earth.  How the earth is moving with respect to the sun, galaxy,etc., is irrelevant.
By saying, "on earth," the frame of reference is specified (the earth), so whatever mass a thing has on earth, (as long as it is not moving with respect to the earth), is its actual rest mass!
In the event you meant absolute rest mass, then the answer is no, because there is no absolute frame of reference. 
