When you first learn some physics of gravity, it is introduced as a force. This is the usual phrasing for Newton’s theory of gravity, whose equations were published in 1687.
But if you then study general relativity (GR) — Einstein’s theory of gravity from 1915 — many textbooks correct you that gravity is not a force, but curvature of spacetime. Congratulations, you have now attained the sophisticated understanding. You have arrived!
…Or have you? When Newtonian gravity is formulated in modern geometric language, it is also interpreted as curvature of space–time. (This uses a manifold with metric(s) and connection, called Newton-Cartan theory.) Hence, the distinction between force and curvature is not about Newton vs Einstein, nor classical vs relativistic, but more about “geometric” formulations vs others.
To throw another spanner in the works, there is a niche approach named teleparallel gravity. Here, gravity is torsion, and there is no curvature at all! The theory uses a different mathematical description than GR, but ends up with the same physical predictions. [Anecdotes: I realised (or re-realised) this interpretation possibility during a lecture at a gravitational waves school at the Physikzentrum in Bad Honnef, Germany in 2019, I think. After the lecture, a Portuguese PhD student and I confirmed with one another what we had heard, and the above consequence. Also in an earlier year, I sat on a streetside kerb in Europe while a young guy (Martin Krššák?) talked passionately to me about teleparallel gravity. We were waiting for some aspect of a conference program to start, perhaps a tour or dinner.]
Yet another concept is inertia. Here I mean a gauge or convention about a “background” velocity or acceleration, for some chosen system at hand. (I am not talking about the more common usage, where inertial motion means geodesic. Regarding another issue, I do not promote an absolute background velocity, as in the aether.) For example, are you sitting still right now, moving at ≈1000 km/h due to Earth’s rotation, or moving at ≈600km/s as determined by the cosmic microwave background? One may ask related questions about acceleration (again, in the sense of an inertial background gauge, not the 4-acceleration 
from the connection in the usual sense). I will write more on this in the future.
The modern view is that gravitation is curvature alone, since only that is physically measurable, locally. (In a ship floating in space, with no windows so you can’t compare yourself to the distant stars, then uniform inertial motion is not detectable, even in principle.) However I have been rethinking the sufficiency of curvature lately. In Newtonian theory, you can add a constant to the gravitational potential, which has no effect on physical predictions. You can also add a term which effects a background/inertial acceleration; this is constant over space but may vary over time. (Malament 2012 
§4.2 has plenty of detail.) I speculate similar gauge choices would apply to relativistic spacetimes too, with appropriate generalisation. I further speculate this is essential for defining gravitational energy, at least in a physical and general way. Another inspiration for going beyond curvature is the Equivalence Principle, a concept which has morphed and diverged quite significantly over the past century. But Einstein’s version of it stressed the role of inertia, as Norton (1985) 
evaluates in detail. I certainly appreciate the modern curvature-only view for what it affirms, just not what it denies.
In conclusion, there is a curious variety of terminology for gravity: force / curvature / torsion / inertia. Personally I continue to conflate “gravity” with “curvature” in communication, but this is mostly just a reflection of standard usage. In particular I do not imply a rejection of the torsion perspective of teleparallel gravity, but merely my unfamiliarity with it. I also wonder if the “force” interpretation might remain useful when frames are specified, including when gravity is described in a similar way to electromagnetism (called gravitomagnetism). As I understand the usage, we say “force” to suggest deviation from the natural trajectory, so the question becomes, which trajectories will one consider natural? Finally, I tentatively suggest an inertial gauge should be absorbed into the concept of “gravity”, in addition to curvature. However, to compromise with the usual terminology, where gravity means curvature, I’ll settle for the hybrid term “inertia–gravity” instead.