Are there any "nice" functions that can take a point from the surface of an n-sphere and map it to a the surface of an (n+1)-sphere?
By "nice", I mean it should be continuous, one-to-one (but not necessarily onto), and cover lots of surface area (not just $f(x) = (x, 0)$).
On
First of all, I do not think this question has anything to do with algebraic topology per se. It appears you are asking for examples topological embeddings $S^{n}\to S^{n+1}$ whose images have positive $n+1$-dimensional Lebesgue measure (this is how I read yours "covers lots of surface area"). You can find some very readable constructions of Jordan curves $J$ in the plane which have positive 2-dimensional Lebesgue measure for instance here. (Note that registering and reading on line at Jstor is free.) Applying the inverse to the stereographic projection, you obtain examples in $S^2$. Multiplying curves $J$ as above by $[0,1]^{n-1}$ and adding "flat top and bottom" you get examples of $n$-dimensional topological spheres in $R^{n+1}$ which have positive $n+1$-dimensional Lebesgue measure.
No. Sard's Theorem. Short version, the image of $\mathbb S^n$ has measure zero in $\mathbb S^{n+1}$ unless the mapping is highly non-differentiable.