If I can solve Sudoku, can I solve the Travelling Salesman Problem (TSP)? If so, how?
For 9x9 Sudoku, no.  It is finite so can be solved in $O(1)$ time.

But if you had a solver for $n^2 \times n^2$ Sudoku, that worked for all $n$ and all possible partial boards, and ran in polynomial time, then yes, that could be used to solve TSP in polynomial time, as completing a $n^2 \times n^2$ Sudoku is NP-complete.  

The proof of NP-completeness works by reducing from some NP-complete problem R to Sudoku; then because R is NP-complete, you can reduce from TSP to R (that follows from the definition of NP-completeness); and chaining those reductions gives you a way to use the Sudoku solver to solve TSP.