If there is no definite speed in the universe, only relative speeds, how does energy increase when velocity approaches the speed of light? Is the concept of energy increasing as it approaches the speed of light based on the fact that this is only true relative to the observer? 
Lets say, there's a scenario where a person in a rocket ship is going past the Earth approaching the speed of light relative to somebody standing on the Earth. From the perspective of the man on the Earth, the energy of the person in the rocket ship & the rocket ship itself have an increased energy compared to their energy at rest, but from the perspective of the person on board the ship, he and the rocket ship have the energy at rest and the Earth has an increased energy compared to it being at rest.
To add on to this, lets say the man on Earth got on his own second spaceship and eventually caught up close to the first spaceship and is approaching the same speed as the first spaceship. Relative to the second spaceship, would the increased energy of the first spaceship gradually lower down to it's energy at rest up until they become the exact speed?
Is this all true?
(I want everyone to understand that I am focusing on the increase of energy due to an object/particle at speeds approaching the speed of light, I already understand how energy increases in Newtonian physics due to the equation of KE.)
 A: Yes, the energy and the increase in energy all depend on your reference frame, but this is NOT special to relativity! The same thing happens in classical mechanics.
I wrote a similar answer to the question, "Can you tell your absolute speed in space?"
Consider the regular Newtonian mechanics equation, $\mathrm{Ke}=\frac{1}{2}mv^2$. If you weigh 50kg, are moving at 0 meters per second and want to accelerate by 1 meter per second, you need $0.5*50*1=25$ joules of energy. If you're moving at $1000$ meters per second (roughly three times the speed of sound in our atmosphere) and you want to accelerate by one meter per second, you need 
$0.5\times 50\times(1001^2-1000^2)=50025$ joules of energy.
This fact does not mean there's a special reference frame, nor that you can tell your velocity through space, nor anything else weird! There's no special relativity here, just everyday classical mechanics. 
You should better understand energy in classical mechanics before you look at energy in special relativity!
A: 
Is the concept of mass increasing as it approaches the speed of light based on the fact that this is only true relative to the observer?

When you say mass increases what you mean is energy increases. You don't want to have a different mass for forces in different directions so the idea of mass changing with speed is now abandoned. You have energy instead of mass increase when speed increases and you have to learn when to use energy versus mass. For instance energy is the source of gravity, not mass.
Mass becomes something that tells you how to balance energy and momentum, if mass is zero equal amounts of both. If mass if not zero then energy exceeds momentum by an amount that depends on mass.
And then there is the point that energy does depend on your frame. Energy depends on your observer, as does momentum.

From the perspective of the man on the Earth, the mass of the person in the rocket ship & the rocket ship itself have an invariant mass that is greater than their rest mass

Absolutely not. You can't say an invariant mass relative to an observer. An invariant thing doesn't depend on the observer. For instance the balance of energy and momentum of a system doesn't depend on the observer, and that is the mass of the system. And it is not the sum of the masses of the parts.
