Consider Problem 8.1-1
I did parts (a) and (b).
For (c), should I solve using either:
Minimum cost method and then method of multipliers?
Vogel method and then method of multipliers?
Is there an easier way to solve (c) (without using software)?
If so, what is it?
The exercise merely asks you to find an optimal solution, so it shouldn't matter how you obtain it, as long as the method you use is correct, and reasonably efficient.
Like other commenters, I had never before come across the Vogel method for obtaining an initial basic feasible solution. While your instructor might regard it as a "better" method than the min cost method, it looks to me like it's also somewhat more complicated. So, if I were doing the problem I'd use either the northwest corner method or the min-cost method. However, if you're comfortable using Vogel's method, you should by all means feel free to use it.
From a web search, it looks like what you know as the multiplier method is the same as what I know as the primal-dual method. Like the Hungarian algorithm for the assignment problem, it's just an adaptation of the primal-dual simplex method which takes advantage of the special structure of the problem. It's therefore sometimes also referred to as a "Hungarian algorithm" even though the transportation problem is a generalisation of the assignment problem. As far as I'm aware, there's no substantially easier way of obtaining an optimal solution.