The challenge here is that something can be topologically a sphere without having the exact geometry of a sphere. For example, if you are walking on the surface of an hourglass shape, it's still topologically a sphere, but if you thread a rope around the 'neck', it won't pull tight.
Instead, consider a torus as topologically identical to a big sphere with an extra loop on it that "warps" you to elsewhere on the sphere (a bit like a gym ball with a handle). We systematically search for loops which don't "pull tight", and test them to see if they're just protrusions or the 'warp' bit of a torus. This comes in two parts:
1: walk in an expanding spiral to search for loops: Fix one end of a rope to the ground, and walk in a circle, leaving a trail of chalk on the ground. Every time you complete a circle, give yourself a bit more rope and try again.
If at any point you cross your own rope, then you've found a loop you need to test. Alternatively, if you end up crossing your own chalk line, then you've found two loops to test (one for each direction on the chalk that you follow back to your home point).
2: test any candidate loops: For this, you basically attempt to spiral inwards, and see if you meet yourself or get 'warped' to another place on the sphere. With different-coloured chalk, trace the outside of the loop all the way around. Then repeatedly trace that edge again but slightly "inside" (that is, away from your home, into the area you haven't walked on yet).
If you end up spiralling yourself into a point, then you've shown that the area enclosed by your loop is topologically flat, and therefore consistent with being on a sphere. You continue searching for more loops to test, until you have covered the whole surface of your world.
Alternatively, if you encounter any lines in your original chalk colour, then that means the loop you found was wrapped around the 'warp' of a torus.
This method should find every possible loop in your world, so if you test them all and don't find a 'warp', then you must be on a sphere.