'$40 a Day' was Rachael's second series, preceeded only by
'30 Minute Meals', and was produced starting in 2002,
according to this page at IMDB, where 'Tasty Travels' is
also shown to have been produced starting in 2005:
Specific dates for the individual shows seem to be unavailable,
as seen on this more expansive listing of her shows, which also
shows '$40 a Day' at the bottom of the timeline, just above
'30 Minute Meals'. The shows are listed numerically, with the
earlier shows at the beginning of each list, so that can give
you some idea of their production range, which was from 2002
to 2003 or later:
'30 Minute Meals' is shown on this IMDB page to have started
production in 2002, as well:
I've personally followed Rachael's various shows and am quite
certain that the majority, if not all of the 22 episodes of
'30 Minute Meals' were shown before '$40 a Day' ever appeared
on television, though their production may have overlapped for
some period in 2002. Production for the 79 episodes of '$40 a
Day' surely continued into 2003 and likely 2004, though a look
at the first link I gave you shows that she was making appearances
on other shows in late 2003 and 2004, so she may have been done
with production of the show at that time.
Also, as noted on this page at IMDB, the 'Tasty Travels' show
is categorized as still being "in production":
Though this may no longer be the case since she began her daily
talk show, 'The Rachael Ray Show', on CBS:
However, it might be possible that she is still producing the
'Tasty Travels' show on weekends.
Ah! TV.com has the original air dates for '$40 a Day', which
shows that the first show originally aired 4/1/2002. The page
lists a total of 77 shows, and shows that the last show
originally aired on 8/12/2005:
Their page for 'Rachael Ray's Tasty Travels' shows that the
first show aired 8/26/2005, and the latest on 9/23/2005:
That should give you a good idea of which episodes are recent
enough that the restaurants named are still in operation, but
keep in mind that Rachael made it a habit to ask people native
to the area for recommendations, which typically led her to
very popular and well-established local hangouts which are much
less likely to have gone out of business or changed ownership.
If the details I've provided aren't precise enough, you can
always contact her by way of the 'Ask Rachael Anything' page
on her personal website:
If anything is unclear, please post a Request for Clarification
before rating this answer.
Additional information may be found from an exploration of
the links resulting from the Google searches outlined below.
Searches done, via Google:
"$40 a day"
"Rachael Ray's Tasty Travels"
rachael ray imdb