How to distinguish between walking on a sphere and walking on a torus?

Get a (two-dimensional) dog and a very long (one-dimensional) leash. Send your dog out exploring, letting the leash play out. When the dog returns, try to pull in the leash. (Meaning, you try to reel in the loop with you and the dog staying put.) On a sphere, the leash can always be pulled in; on a torus, sometimes it can't be.

(See homotopy.)

The Gaussian Curvature is an example of an intrinsic curvature, i.e. it is detectable to the "inhabitants" of the surface. The Gauss-Bonnet Theorem gives a connection between the Gaussian Curvature $K$ and the Euler Characteristic $\chi$. For a smooth manifold $M$ without boundary: $$\int_M K~\mathrm{d}\mu = 2\pi \chi(M)$$ The Euler Characteristic and the genus of the surface are connected by $\chi(M) = 2-2g$. A sphere has genus zero and so $\chi(S^2) = 2$, while a torus has genus one and so $\chi(T)=0$.

You could, as the ordinance survey people do, choose triangulation points on your surface, measure the Gaussian Curvature at those points and then use this to approximate the above integral.

Travel a lot and depict a map of the world. Then try give a colour to every state in your map, in order that neighbours have different colours. If you need more than four colours, you are on a torus.

This is just a reformulation of @Fly by Night's solution, since the chromatic number depends on the genus.

In a more deterministic way, on a torus you can embed a $K_5$, i.e. you can find $5$ points $A_1,\ldots A_5$ such that there exist $10$ non-intersecting paths from $A_i$ to $A_j$, on a sphere you cannot.

As an alternative, given two distinct points $A$ and $B$ on the surface, you can draw the locus of equidistant points (with respect to the geodetic distance) from $A$ and $B$. If such a locus has two connected components, you are on a torus.

Another possibility is to "comb" the surface. If you are able to, you are on a torus. And I bet that there is a plethora of opportunities given by the Borsuk-Ulam theorem, in general. For example, on a torus the wind (as a continuous vector field) can blow with a non null intensity in every point, on a sphere it cannot.

Or try to draw many concentric circles. If you are on a torus, sooner or later one of these circles must intersect itself.

And thanks to Giovanni Barbarino, on a toric surface there is always a point with a zero gravity, so there are some issues in building homes nearby.