Is Newton's first law something real or a mathematical formalism? Why do objects always 'tend' to move in straight lines? How come, everytime I see a curved path that an object takes, I can always say that the object tends to move in a straight line over 'small' distances, but as you take into account the curvature of the path, a force acting on the particle appears. I mean, I can always take a small enough portion of the curve, zoom in enough, and conclude that the object is moving in a straight line, but then as I zoom out I find out that a force is acting on the particle. The force of gravity is everywhere and, no matter how weak it is, it will make the particle take a path which is different from a straight line. This is my question: since particles are, in reality, never moving in straight lines, is Newton's first law a mathematical formalism or some true property of material objects? 
 A: Nice question! The answer to this depends on the version of Newton's first law you use.
In the Principia, the statement of the first law, as translated by Machin, is:

Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

This is immediately followed by a series of examples:

Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.

Of the three examples, not one involves motion in a straight line! Since the first law is stated in the Principia in words rather than equations, there's a lot of room for ambiguity. Keep in mind as well that scientists reading the Principia in that era didn't know calculus, and vectors weren't invented until centuries later. Newton had to write in language his contemporaries would understand, even if it was at the cost of precision.
There are many different ways in which the first law has been stated over the years, as described in this question: History of interpretation of Newton's first law .
You can modify it to be a statement that if you choose a specific axis $x$, then the absence of any forces in the $x$ direction gives $dv_x/dt=0$ at that instant in time. This is probably the interpretation that's most directly suggested by Newton's three examples.
You can modify it to be a statement about objects that are acted on by zero total force.
As described in the other question, it's now popular (probably due to the influence of the analysis in Mach 1919) to describe it as a statement about the existence of inertial frames.
Gravity does present some unique issues, since it's a long-range force and can't be shielded against. Mach 1919 gave a very thorough and insightful critique of the logical basis of Newton's laws. Here is my own presentation of the question of what the first law really means and some experimental tests. In general relativity, we define a free-falling frame as an inertial frame, so that the motion of a projectile is defined to be "straight."
Ernst Mach, "The Science Of Mechanics," 1919, http://archive.org/details/scienceofmechani005860mbp
A: Your premise appears to be that imperfections in the physical world can invalidate precise mathematical statements. However, there is a difference between a statement not applying in detail and a statement being entirely invalid.
Sure, you can always look closely enough and find forces acting on an object. Similarly, if you have a gas someone claims is at $300\ \mathrm{K}$ you could always look close enough and find deviations. But that doesn't mean the concept of "$300\ \mathrm{K}$" isn't intimately tied to physical objects. The property

ability to reach $300\ \mathrm{K}$,

or more pedantically

having continuous behavior in a neighborhood of $300\ \mathrm{K}$, such that any experimental prediction made assuming $300\ \mathrm{K}$ can be obtained to arbitrary accuracy by being sufficiently close to $300\ \mathrm{K}$,

very much belongs to real objects in the real world, even if none of them in practice have the property

currently at exactly $300\ \mathrm{K}$.

As I see it, then, the law of inertia can be seen to hold in two different ways:


*

*A statement about inertial objects can be viewed as a limit. That is, "real" objects' behavior asymptotically approaches that predicted by Newton's first law as extraneous forces are removed from the system. In this sense then there is meaning to the law and it is not arbitrary - any other choice of limiting behavior would contradict reality as we observe it.

*Working in the other direction, the law provides a starting point for investigating forces. If there were no baseline, so to speak, for objects' motion, how then could we even go about discussing forces and accelerations? It would be difficult (perhaps not impossible, but still rather unsatisfying I believe) to make a cogent metaphysical argument in which objects are affected by forces yet a hypothetical object unaffected by forces has an undefined motion. This goes back to the existence of inertial frames Ben Crowell mentioned.
A: An invaluable read for you would be Richard Feynman the characteristic of physical law. He explains in his inimitable manner that laws can only ever be described by mathematics. Then experimentation tries to reveal imperfections in the model described by the math. As with Einstein bringing relativity in to the previous view of Newtonian mechanics, one can never rule out an observation in nature may one day falsify a truth that you currently hold dear.
A: 
This is my question: since particles are, in reality, never moving in straight lines, is Newton's first law a mathematical formalism or some true property of material objects?

You assume that particles never move in straight lines. This cannot be proven and is not necessary in physics. But even if you assume it, then Newton's 1st law states that these particles not moving in straight lines are acted upon by other bodies. It is a statement about existing situation, it is non-mathematical, so I think it is a physical law.
In Newton's view, it is not a property of the material objects only. The important part is that the motion is to be observed with respect to absolute space, or in the view common today, inertial frame.
A: The trajectory of an object you consider is dependent on the scale on which you track its motion. For objects over small speeds or distances, the necessity to correct the trajectory to account the curvature of the earth or the gravity of the moon doesn't exist. However, if you consider longer distances, longer scales or higher velocities, the trajectories are more "sensitive" to smaller changes i.e., seemingly smaller things (like the curvature of the earth or the gravity pull of the moon) have a larger influence on things of a.) higher mass, b.)higher velocity c.)processes on a larger time scale.
So, they don't actually move in a straight line, but for Newtonian mechanics, you can assume they do and you'd not be measurably wrong.
A: There are at least two distinct things you can do with a physical theory, one is to use is it to understand the universe, the other is to make a testable prediction that could aim to falsify the theory.
There is some potential overlap, for instance, to make a testable predictable, you have to be able to compute something accurately enough to have a prediction, so the theory has to postulate its predictions in terms of something well-understood enough to allow predictions.
But since understanding is so personal, I'll now focus on Newton's Laws as they are used to make predictions and falsify other theories and laws.  Which is what I think the laws are about, so not merely (or really at all) mathematical.
Newton's laws of motion first require some laws of mass and forces (these are terms that come up in Newton's laws).
Newton's law of mass were about how mass depended on say size or density (less you think that is tautological, relative size and relative density are things you measure, so mass is a thing proportional to both).
Then there are Force Laws (there are many, ones for gravity, ones for springs, etc.)
Newton's third law of motion then constrains what force laws you consider (effectively you only use/consider force laws that conserve momentum).  So this law isn't math, it's telling you to reject certain force laws even before you do any measurements.
Newton's second law of motion turns these force laws into predictions about motion, thus allowing the force laws to be tested, not just eliminated for violating conservation of momentum. This works because he postulates that we can test force laws by using calculus and then looking at the prediction from solutions to second order differential equations.  This one can help you to understand a force law by understanding solutions to differential equations.  So this law isn't math, it's telling you to reject certain force laws after you do some calculations (predictions) and measurements.
Newton's first law of motion then excludes certain solutions that the second law allowed. I'm not saying that historically Newton knew this, but it is possible (see Nonuniqueness in the solutions of Newton’s equation of motion by Abhishek Dhar Am. J. Phys. 61, 58 (1993); http://dx.doi.org/10.1119/1.17411 ) to have solutions to F=ma that violate Newton's first law. So adding the first law says to throw out those solutions.
In summary: the third law constrains the forces to consider, the second makes predictions so you can test the force laws, and the first constrains the (too many?) solutions that the second law allows. They all have a purpose, they all do something.
And you need to first have laws of mass and/or laws of forces before any of Newton's laws of motion mean anything.  The purpose of the laws of motion isn't to be mathematics, it is to falsify laws of force and mass.  That's what they are, that's what they do.
