A few more things I can mention. The stress-energy tensor first makes its appearance in special relativity (henceforth SR), so it's needed to understand mechanical problems in SR. A typical SR problem that we've seen on PF that requires the stress-energy tensor would be calculating the angular momentum of an idealized rotating hoop. Be warned, we've also seen a lot of confusion on the part by people not familiar with tensors who try to understand this problem - it's not a particular easy problem to get right without the right methods.
Pedagogically, I would suggest learning about tensors first, before attacking general relativity. It's possible to understand about tensors in isolation, I suppose, but I'd generally recommend a physical application. Applied tensors are usually introduced in electromagnetism. Understanding why the electric field and the magnetic field are combined to form one larger tensor, the electromagnetic field tensor or Faraday tensor, is probably a good first introduction to the use of tensors in physics, and aids in understanding the interrelationships between seemingly different ideas. Understanding this point will aid one in understanding why stress (including pressure) and density are related the way they are in special relativity.
The relationship between stress (including pressure) and density is unexpected, but the principles unifying them into a lager entity are similar to the principles that unify the electric and magnetic fields in electromagnetism. The tensor methods unify the electric and magnetic field into the Faraday tensor - similarly they unify stress, energy, and momentum into the stress-energy tensor.
electromagnetism (EM).
A prerequisite of understanding tensors in general is partial differential equations, henceforth PDE's. PDE's can also be taught in isolation, but are frequently bundled in an applied context, typically Maxwell's equations. The first glimpse one has of Maxwell's equations is presented without tensors. At the graduate level, one introduces tensors and revisits the problem of Maxwell's equations, learning new and powerful techniques.
As a side bonus, one will gain a better understanding of covariance. Tensors transform partial differential equations in a manner that's by definition unaffected by coordinate transformations and/or changes in "frames of reference". I this is a unaswered question for a lot of people, though the mathematical requirements to appreciate how tensors answer this question can be daunting. Tensor methods also play well with Lagrangian methods, as those methods are also about expressing physics in "generalized coordinates". Lagrangian methods are also stressed at the graduate level, though they may be first introduced earlier. It is not essential to learn about Lagrangian mechanics before tackling tensors, however, it can be left until later.