Electric Vs Magnetic Field Lines Let us consider this:

One part of the force between moving charges we call the magnetic force. It is really one aspect of an electrical effect.
  — Richard Feynman

From relativity, this is infact true. Then why electric field lines are straight, pointing radially outwards, and magnetic field lines are in the form of a ring or a loop?
i)In other words why are they both different?
I should be able to see them the same way, no matter the frame of reference, right?

Now let me assume that the field lines  are the way they are.
So let me get on my trolley and move along with the charged particle. I will see the charge getting repelled by electrostatic force. And I see the electric field lines(radially outwards)
Now I slowly decrease my speed....and finally come to rest.
At rest I will see the charge moving, as well as getting repelled by magnetic force. And now I will see magnetic field lines in the form of concentric circles.
ii)During my journey how will I see the  lines getting transformed into loops?
P.S: The entire question is revolving around the phenomenon of a charged particle moving alongside a current carrying wire
Edit:From relativity, is magnetic field just a moving electric field? Or it just relates the forces? 
 A: 
I should be able to see them the same way, no matter the frame of reference, right?

No.  When you change to a boosted reference frame which has velocity $\mathbf v$ with respect to the old one, the electric and magnetic fields transform as follows:
$$\mathbf E_\parallel' = \mathbf E_\parallel$$
$$\mathbf B_\parallel' = \mathbf B_\parallel$$
$$\mathbf E_\perp' = \gamma(\mathbf E_\perp + \mathbf v \times \mathbf B)$$
$$\mathbf B_\perp' = \gamma(\mathbf B_\perp - \frac{1}{c^2}\mathbf v \times \mathbf E)$$
where $\parallel$ and $\perp$ indicate the directions parallel to and perpendicular to the boost velocity $\mathbf v$, and $\gamma \equiv \frac{1}{\sqrt{1-v^2/c^2}}$ is the Lorentz factor.


From relativity, is magnetic field just a moving electric field? Or it just relates the forces?

The magnetic field and electric field are two aspects of one physical entity which we call the electromagnetic field.  Different observers see different combinations of electric and magnetic fields, but they combine in such a way that all observers will agree on how a charged particle will move, in the sense that the particle's coordinates in one frame are related to its coordinates in another frame by the appropriate Lorentz transformation.

From a comment,

Can you please provide additional details to your answer about electromagnetic field and how it changes it's form- from purely magnetic to purely electrostatic ?

If the EM field is purely magnetic in one reference frame, then you won't be able to find a reference frame in which it's purely electric.  The reason is that even though $\mathbf E$ and $\mathbf B$ are not Lorentz-invariant (which means they change when you change reference frames), the quantity $\mathbf B^2 - \mathbf E^2/c^2$ is Lorentz invariant.  In particular, if the field is ever purely magnetic then this quantity is positive; in order for it to be purely electric, this quantity would have to change sign, but it remains constant during Lorentz boosts.
A: I have read on this website that the electromagnetic field is an entity which changes its appearance when viewed from a different moving coordinate system, but both components are real.  Keep in mind that magnetic field lines are a mathematical construct designed to represent a complex situation.  To find a magnetic force you have to go through two “right hand rules”. One to define the direction of the field, and another to find the direction of a force. The actual force between two parallel current carrying wires is either attraction or repulsion.
