First let's be perfectly clear on one point: There is no evidence whatsoever that wormholes exist in nature; in fact all evidence points toward their existence as being impossible. Wormholes require the existence of a new class of matter that we have no reason to believe exists. Furthermore, the existence of a wormhole would undermine causality in the universe as we know it.
Now, supposing you found some magical material that lets you create a wormhole. The topology (loosely, the global connectivity) of spacetime would be nontrivial, but the geometry (the local sense of size and direction) would not be so exotic. Indeed, general relativity is predicated on the geometrical notion that spacetime is a smooth manifold. In other words, it locally looks like the flat spacetime of special relativity. In still other words, around any point we can always find a small enough region and a suitable change of coordinates such that physics looks arbitrarily close to special relativity in the region, with no gravity to speak of.
As long as your wormhole warped spacetime gently enough, regions the size of a person or a starship or whatever can be taken to be small enough in the above sense. That is, the internal tidal forces stretching or shearing you as you traverse the wormhole can be made arbitrarily small with enough tinkering with the wormhole's parameters. This is the equivalence principle at work -- inside a small enough free-falling box, you cannot even in principle tell what gravitational fields might be around you, warping spacetime.
