This is a bit complicated. Of course, it's right that "force" is a difficult concept in relativistic physics, and you should get used to the field concept. In the case of gravity many people stress the geometric interpretation, i.e., gravity as the dynamically evolving pseudo-Riemann space-time manifold. Nevertheless it's (at least for me as a theoretical nuclear/particle physicist) simpler to think about gravity as a relativistic field with the pseudometric as gauge fields.
Of course, operationally, as any field also the gravitational field is defined by the motion of test particles, and indeed that's how most textbooks on the subject starts. As an introduction I like most Landau/Lifshitz vol. II. The development of general relativity starts with the motion of test particles in the gravitational field. The unique characterization of gravity in comparison to the other fundamental interactions (described as the strong and electroweak interactions in the Standard Model) is that all test particles with the same initial conditions move along the same trajectory in the gravitational field, no matter what's their (invariant) mass or their material. As it turns out this leads to a description of the motion of such testparticles along geodesics in a curved spacetime manifold, and that's where the geometric reinterpretation of the gravitational gauge field as Riemannian pseudometric comes from. The equation of motion, however indeed is not so much different from the relativistic motion of a particle in an external field, and in this sense for this case you can indeed think in terms of a locally acting external force (as in electromagnetics, where you can think of this situation of a charged particle moving in an external electromagnetic field as the motion of the particle subject to the electromagnetic Lorentz force).
Gravitational waves are also pretty much analogous to electromagnetic waves. The main difference is that as a non-abelian gauge theory gravity is a self-interacting field, i.e., the Einstein equations of motion are non-linear even without matter present. Gravitational waves were, however, predicted by Einstein working with the linearized field equations, and there it's pretty much just a wave equation for the assumed to be small deviation of the gravitational field from the "vacuum solution", i.e., the Minkowski metric. The manifestation of a gravitational wave to test particles is similar to that of the electromagnetic wave field on test charges. The only difference is that the multipole expansion of gravitational waves starts with the quadrupole term, because the gravitational field is a spin-2 field rather than a spin-1 field (both massless).
So you can very well think about the action of the gravitational wave field on matter (among them the LIGO detectors) as the action of a force, and this is indeed use to detect them with these very sensitive interferometers.