I will go backwards, hopefully avoiding any emesis on the way...
There is no textbook in English that I know of which teaches Vedic. Like Homeric Greek, Vedic Sanskrit is an earlier form of the language mixed with an unnaturally lush collection of alternative grammatical forms. Actually, there's some reason to believe that Vedic and the ancestor of Classical Sanskrit are siblings, rather than that Vedic is the parent of Classical. That said, you have to start with Classical.
Everyone still uses Lanman's Sanskrit Reader
to start. All over the English speaking world are people who can probably still recite the opening shloka's of the Nala story, even if they can remember nothing else of the Sanskrit they once studied. It starts off with Classical, but has a hefty Vedic section, too. Then MacDonell's Vedic Reader
would be a good follow-up. He also has a grammar for both classical and Vedic.
Whitney's Sanskrit Grammar
is also still a standard tool, but Jan Gonda's A Concise Elementary Grammar of the Sanskrit Language
is worth having. When I studied Sanskrit, that was the book the TA always had with her.
Except Gonda's, all the books I've mentioned above can be easily found in cheap reprints of varying quality. Take care that any copy of the Vedic Reader
you order actually has the Sanskrit text, too. The front matter and translations have floated around the net freely for quite a while.
The Perry Sanskrit Primer
— actually a translation of a German work — is a grand old book that can still be found in used bookstores without much effort. When I took Sanskrit the textbook was A Rapid Sanskrit Method
which, while recent, shouldn't cause a classicist too much nausea. Devavanipravesika
is still used in college courses, and again though recent should present a face recognizable to a classicist (though, I've not actually paged through a copy in more than a decade).