The basic equations you need for this are called the geodesic equations. The most direct form is when your spaceship has no thrust, then one can write (
https://en.wikipedia.org/wiki/Solving_the_geodesic_equations)
$$\sum_{b,c=0..3} \frac{d^2x^a}{ds^2} + \Gamma^{a}{}_{bc}\frac{dx^b}{ds}\frac{dx^c}{ds} = 0$$
Here the ##x^a## are coordinates, for instance ##x^0 = t, x^1 = x, x^2=y, x^3=z##. The equations are in parametric form, so we are solving for a solution of the form
$$t(s), x(s), y(s), z(s)$$
or in the original notation
$$x^0(s), x^1(s), x^2(s), x^3(s)$$
There are 4 equations, corresponding to values for a of 0, 1,2, and 3. b and c are dummy indices, which one sums over, so there are 16 terms if you write out all the terms.
The ##\Gamma## coefficients are called "Christoffel symbols", and they can be calculated from the metric. The metric describes the space-time, you can use the plain Schwarzschild metric for starters, which would be a non-rotating black hole. If you want to use cartesian-like coordinates (x,y,z) rather than (r, ##\theta##, ##\phi##) you probably want to use not regular Schwarzschild coordinates, but isotropic coordinates. But it's much easier to stick with regular Schwarzschild coordinates and use ##r, \theta, \phi##.
In GR you can use any coordinates you like, which is a powerful feature of the theory that can be confusing - it makes no physical difference (just as it makes no difference whether you use Cartesian or polar coordinates in a freshman physics problem), but you have the responsibility of choosing coordinates that mean something to you, and you have the opportunity to confuse yourself over such issues. It doesn't sound like a major problem, but people often think about problems in terms which require specific features of the coordinates (like the ability to find distances by subtracting coordinates), which hinders their undertanding of the significance of a solution written in coordinates that do not have these special features. A classic example of this is to interpret "r" in Schwarzschild coordinates as being a radius- the issue is that subtracting r values does not give you the distance.
The Christoffel symbols are calculated from the metric coefficients, which describe the space-time. See for instance
https://en.wikipedia.org/wiki/Christoffel_symbols
$$\Gamma_{abc} = \left(\frac{\partial g_{ca}}{\partial x^b} + \frac{\partial g_{cb}}{\partial x^a} - \frac{\partial g_{ab}}{\partial x^c} \right)$$
Unfortunately, you need to learn how to raise the index, as you actually need ##\Gamma^a{}_{bc}##, not ##\Gamma_{abc}##. Wiki has the forumla, but it uses tensor notation. And it'd be a bit long to explain.
For an overview of putting this all together to find an unpowered orbit, you might look at
http://www.fourmilab.ch/gravitation/orbits/, "Orbits in Strongly Curved Space-time". The treatment follows MTW's treatment in "Gravitation" fairly closely. It has a java applet, which may be in somewhat of a state of disrepair, last time I tried to use it it raised a bunch of security warnings.
I haven't discussed in detail how to add thrust, basically rather than set the geodesic equation to zero, you'd set it to the thrust value. But I think a good first goal would be to write and solve the geodesic equations for an unpowered test particle orbit in a Schwarzschild space-time.
This is already too long, and it's barely started. But I hope it's of some help.
Sean Caroll's lecture notes on GR might be a good source to replace MTW as a formal reference - they're available online, but not an easy read.
https://www.preposterousuniverse.com/grnotes/. But they might be rather technical.
