I'm not going to answer this question in complete detail, since it might be homework, but here are some tips on how to proceed.
First, decide if there's an exact solution available. If it's linear, you have a good chance. The class of nonlinear ODE for which there is an exact solution can seem disappointingly small, but it's worth knowing the tricks (maybe it's exact, maybe you can use the Frobenius method, perhaps there is a change of variables that simplifies things, etc.). If stumped, another tool is a Computer Algebra System.
If you decide there isn't an exact solution, you can choose a numerical method to compute an approximate solution. This depends on the boundary/initial conditions. Whether you have an "initial value problem" (information about $y$ and its derivatives at a single point in the domain) or a "boundary value problem" (information about $y$ and its derivatives at two separate points in the domain) makes a lot of difference.
Based on the comment below, you are interested in an initial value problem, which for concreteness we'll say is
$ y''(t) = 1/y - 1$, $y(0) = y_0$, $y'(0) = 0$. Now one can choose freely from the bestiary of ODE solvers available. Note that if you can show that the solution stays bounded away from $y=0$, things will behave nicely. The form of this second order ODE makes it possible to use second-order methods like Verlet methods, you can pick an old workhorse like Runge-Kutta-Fehlberg(aka ode45 from MATLAB), or, if it seems likely that the solution will approach $y=0$, an implicit and/or stiff solver like a . Note that by introducing a new variable $v \doteq dy/dt$, you can write the second order ODE as a system of two first order ODEs.