Advantages of Lagrangian Mechanics over Newtonian Mechanics
It is necessary to study Newtonian mechanics to truly understand Lagrangian mechanics since its underlying foundation is Newtonian mechanics. It is essentially a different formulation of the same thing. In a way when doing Lagrangian mechanics you are still doing Newtonian mechanics just in the way of energy. For example, under Lagrangian mechanics, say we have a particle with some kinetic energy, ${T=\frac{1}{2}m\dot{q}^{2}}$, that is in a gravitational field, $V=mgq$. Our Lagrangian is defined as $L=T-V$, so using the Euler-Lagrange equation, ${\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}-\frac{\partial L}{\partial q}=0}$, we would get $m\ddot{q}+mg=0$, which you can see is just Newton's usual sum of forces telling us in this case that the acceleration, $\ddot{q}$, here is just due to gravitational acceleration, $g$.
While this may seem like a convoluted way of getting to the same thing, you can use a different example to solve for a much more complicated system like a double pendulum [pdf link] by both methods to drive the point of why Lagrangian mechanics is the method of choice.
You can see that Lagrange mechanics provides a much more elegant and direct way of solving these complicated systems especially if you start adding in damping or driving mechanisms.
One of the attractive aspects of Lagrangian mechanics is that it can solve systems much easier and quicker than would be by doing the way of Newtonian mechanics. In Newtonian mechanics for example, one must explicitly account for constraints. However, constraints can be bypassed in Lagrangian mechanics. You can also modify the Lagrange equations pretty easily as well if you want to account for something like a driving or dissipation forces.
No, I would highly recommend studying Newtonian mechanics before Lagrangian mechanics. While, yes it is 'possible' to learn about Lagrangian mechanics before Newtonian, a lot of intuition would be lost beginning with one instead of the other which will, in the long run, do no more than harm you or, at best, possibly confuse you. But there are, indeed, many advantages to this formalism.
Though ERK's answer gives some good reasons (i.e. simplicity of solutions and such), I think the solution glosses over a crucial part of Lagrangian Mechanics (which I'll just post for completeness): it allows us to work in generalized coordinates and are completely invariant to them.
Whereas the Newtonian formulation requires explicit rewriting of its laws in order to deal with arbitrary coordinate systems, the Lagrangian formulation (which is, if I recall correctly, slightly weaker than the original Newtonian formulation) in turn, allows us to deal with arbitrary coordinate systems on spaces which suit our problem.
A simple example comes from rewriting each formulation in polar (2D) coordinates. Consider rewriting the definition of force in two dimensions (assuming $m=1$): $$ (\ddot r - r\dot\theta^2)\hat e_r + (r\ddot\theta +2\dot r \dot \theta)\hat e_\theta = a = -\nabla U = \frac{\hat e_\theta}{r}\partial_\theta U + \hat e_r \partial_r U $$
where most of the terms in the LHS emerge from differentiation of the basis vectors (as they change at each point). On the other hand, the Lagrangian expressions retain their usual form: $$ \begin{align} \partial_r L &= \frac{d}{dt}\partial_\dot r L\\ \partial_\theta L &= \frac{d}{dt}\partial_\dot \theta L \end{align} $$
and all we have to do is rewrite the forms of the kinetic/potential energy in polar coordinates (which you often have if you're using this method to exploit symmetries in the problem). This, in particular means that constraints can be enforced by choosing appropriate coordinate systems which suit the given problem rather than by explicitly writing the constraints and solving for them (as we would often have to do in Newton's equations).
Additionally, there's a lot of niceties that can be proven directly from the Lagrangian and its corresponding action (which is the underlying reason for these particular invariances), most notably Noether's Theorem which states that every Lie symmetry of the action corresponds to a conservation law; for example, if the Lagrangian of a particular system is invariant under infinitesimal translations in time, that system's total energy is conserved.
It's true that these kind of theorems can (theoretically) be proven directly from Newton's laws (since they are, in some odd sense, a consequence of them), but the symmetries of the laws are not readily apparent until recast in this formulation.
Possibly related questions: What is the difference between Newtonian and Lagrangian mechanics in a nutshell?, What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)
You should study Newtonian mechanics before Lagrangian mechanics because Newtonian mechanics is more general than Lagrangian mechanics. In other words, while whenever a system allows a Lagrangian formulation it also allows a Newtonian formulation, the converse is not true; the quintessential case is dynamics in the presence of dissipative forces. Lagrangian dynamics allow a very restricted class of dissipative forces to be treated, i.e. those that depend on velocity only, see for instance this online discussion. But the most general case (think for instance of a coin falling in a stratified atmosphere, spinning about its axis but with its symmetry axis not parallel to its instantaneous speed) is completely outside the reach of Lagrangian dynamics.
If you think this example is contrived or far-fetched, think of an airplane wing moving inside a turbulent air layer, and of the importance of the computation of the lift and drag coefficients.
At the same time, you may ask yourself why we then persevere in studying Lagrangian mechanics, if its field is obviously restricted to forces which can be derived from a potential. There are many reasons:
All fundamental forces in nature can be derived from a potential, of sorts; while we are interested in keeping our airplanes afloat, it is also true that electromagnetism, gravity, weak and strong interactions can all be derived from a Lagrangian;
Lagrangians make the derivation of the equations of motion in generalized coordinates immediate, instead of projecting things onto axis, and getting lost in the details of geometry;
Lagrangians make the discussion of invariance principles easy, by exposing the connection between the symmetries of the Lagrangian with the existence of conserved quantities (Noether's theorem), and by making the discussion of symmetries in the Lagrangian trivial. Consider, as an example, the derivation of the conserved quantity for the motion of a point particle in the field generated by an infinite helix: from the symmetry of the Lagrangian it is easy to show what the conserved quantity is (it is one of the first exercises in Landau and Lifshitz; vol 1 Mechanics), while try to do the same in Newtonian mechanics. Notice that I have not specified which kind of field is generated by the helix, because the conserved quantity is always the same, irrespective of the nature of the field (provided it can be derived from a potential).
A Lagrangian is associated with the concept of a minimum and, while the nature of this minimum per se is not extremely important, it leads to numerical approximation schemes (the so-called relaxation methods) which are sometimes our only, and very often our best approach to a concrete problem.
If you will allow me a foray outside Classical Mechanics, a Lagrangian treatment of a problem allows a powerful analogy with QM's non-commuting operators, and the introduction of commutators and anti-commutators, which was a key step in the development of QM.