Why don't we constantly experience a mass dilation? So according to the principle of relativity there is no preferred reference frame, and thus from the reference frame of a fast-moving particle all objects would be moving  close to the speed of light. The mass dilation transformation says that objects gain relativistic mass when they approach the speed of light, so do all objects have a high relativistic mass? I know I'm misunderstanding Special Relativity, but how can mass differ between reference frames? 
 A: That is why in special relativity   the concept of invariant mass is the relevant one, not the relative mass between moving systems , but the mass of the object at its rest frame.
Relativistic mass is mathematical concept that demonstrates the mass/energy equivalence but is a confusing concept. It is called "mass" because when studied from another frame of reference it reacts to changes in momentum ( force=dp/dt) as the Newtonian kinematics  mass :

In physics, mass is a property of a physical body. It is the measure of an object's resistance to acceleration (a change in its state of motion) when a force is applied.

you ask:

when they approach the speed of light, so do all objects have a high relativistic mass?

All objects have their fixed invariant mass in their own system. The mathematics of special relativity is such that this mass measure is invariant even when measured on moving systems .  We measure the invariant mass of elementary particles with great accuracy using relativistic kinematics on moving systems. These elementary particles move with respect to us usually close to the velocity of light. We do not calculate their relativistic mass because it is not an invariant, it changes according to 

where m_0 is the invariant mass we measure and identify the particle.

Why don't we constantly experience a mass dilation?

The particle experiences no changes in its invariant mass due to the existence of fast moving other particles. It will simply observe other particles as having high relativistic mass with respect to it.
 At the center of mass system of the particle, if it could measure, it would observe everything as having an enormous relativistic mass.
Objects observed from a moving system may have a very large relativistic mass. The relativistic mass is not an invariant measure of special relativity but depends on the relative velocities between the observing system and the observed system.
A: Energy, in particular kinetic energy, and momentum depends on the reference frame. In special relativity, when a body moves close to the speed of light $c$ with speed v, its momentum is given by $$p=\frac {m_0}{\sqrt{1-(v/c)^2}}v$$ where $m_0$ is the rest mass of the body and its total energy is given by $$E=\frac {m_0 c^2} {\sqrt{1-(v/c)^2}}$$ Thus both the momentum and the total energy of the body approach infinity when $v \to c$. When you interpret in these formulae the term $$ \frac {m_0} {\sqrt{1-(v/c)^2}}$$ as the "relativistic mass" this mass goes to infinity when $v \to c$ explaining the impossibility to accelerate the body to $c$. This is equivalent to stating that the total energy goes to infinity. This "relativistic mass" is obviously reference frame dependent, just like length and time intervals, in contrast to the rest mass, which is frame independent. Thus if you move close to the speed of light to all other bodies, all these bodies gain energy and thus "relativistic mass" relative to you. 
A: 
so do all objects have a high relativistic mass?

No, because there are very few objects moving at speeds close to the speed of light due to Aristotle's principle.

according to the principle of relativity there is no preferred
  reference frame

In special relativity there is not. In general relativity, there is.
