Using Lagrangian mechanics to solve a motion of one particle

There isn't much to solve actually. The Euler-Lagrange equation is $$ frac{d}{dt}frac{partial L(r,dot r,t)}{partial dot r_i} = frac{partial L(r,dot r,t )}{partial r_i} $$ where $r$ stands for the collection of all coordinates $r_i in r$. In your case $r=(x,y,z$). The Lagrange function for a free particle is simply $$ L(r,dot r,t) = frac{1}{2}m(dot x^2+ dot y^2+ dot z ^2 ) = L(dot r) $$ which is a special case of a Lagrange function that does not depend on $dot r $ and $t$.

The Euler-Lagrang equation tells us now how to derive the differential equations that need to be solved to obtain the trajectories of the system.

In the derivatives $frac{partial L}{partial r}$ and $frac{partial L}{partial dot r}$ we treat $r$ and $dot r$ like independent variables following normal differentiation rules as you know them from normal functions. This leads to

$$ frac{partial L(dot r)}{partial x} = frac{partial }{partial x}frac{1}{2}m(dot x^2+ dot y^2+ dot z ^2 ) = 0 frac{partial L(dot r)}{partial y} = frac{partial }{partial y}frac{1}{2}m(dot x^2+ dot y^2+ dot z ^2 ) = 0 frac{partial L(dot r)}{partial z} = frac{partial }{partial z}frac{1}{2}m(dot x^2+ dot y^2+ dot z ^2 ) = 0 $$ The derivatives with respect to $dot x, dot y, dot z$ are $$ frac{partial L(dot r)}{partial dot x} = frac{partial }{partial dot x}frac{1}{2}m(dot x^2+ dot y^2+ dot z ^2 ) = mdot x frac{partial L(dot r)}{partial dot y} = frac{partial }{partial dot y}frac{1}{2}m(dot x^2+ dot y^2+ dot z ^2 ) = mdot y frac{partial L(dot r)}{partial dot z} = frac{partial }{partial dot z}frac{1}{2}m(dot x^2+ dot y^2+ dot z ^2 ) = m dot z $$

Now we can plug in the derivatives into the Euler-Lagrange equation, $$begin{aligned} frac{d}{dt}frac{partial L(dot r)}{partial dot x} &= frac{partial L(dot r)}{partial x} frac{d}{dt}mdot x &= 0 end{aligned}$$ which leads to $$ mddot x = 0 $$

This is the same as $$ma = F$$ with $F=0$, as expected for a free particle.

During the formalism, we sort of use $x, dot x$ with two different meanings. In the derivatives they are simply independent variables of the Lagrange function. So treat the Lagrange function like you would treat any other function with different variables, for example $$ f(a,b) = b frac{partial f}{partial a} = 0 $$ Once you are done with the derivatives, you substitute the independent variable $x$ with $x(t)$ before evaluating the time derivative to obtain the differential equation for the trajectory $x(t)$.