How large is the set of all Turing machines? How large is the set of all Turing machines? I am confident it is infinitely large, but what kind of infinitely large is its size?

Tac-Tics's answer is almost correct, but the "naive program" argument is unnecessary and possibly incorrect (as per my comment).
The observation that Turing machines can be described by finite strings in a finite alphabet is sufficient: by mapping the letters of the alphabet to integers in some base (e.g. binary or, as Turing does, decimal without the use of digits 8 and 9), you can map all programs to integers. Thus the set of all (valid or non-valid)1 machines is mappable to a subset of the integers; thus both the set of all machines and the (smaller) set of all valid machines are countable.
1: By "valid," I mean either Turing's "circle-free" concept or the later equivalent-but-inverted "halting" concept (introduced when Turing's work was reframed as the "halting problem").
