Hello Samantha,
If I may, here is a quick course:
A quick trip to
www.handgunlaw.us will give you a good picture of which states are "may issue" (mostly political issue or insanely difficult requirements) and "shall issue" (must issue to any qualified citizen), which states only honor "home state" permits (CO, MI, SC, NH, ME, FL) and a state by state accounting of whose permits each state honors and which states honor that state's permit and whether or not a given state honors non-resident permits. This site will also let you compile a list of states that only issue to residents (example; WI, MN, CO, etc.) You will also learn what each state requires for a permit. I have found this to be a very good source as the webmasters provide links to the permit issuing authority in each State.
The UT, MN, FL, and AZ permits are the most widely accepted, and thus most popular, but MN does not issue to non-residents. UT, FL, and AZ will issue to non-residents and permits can be applied for by mail. FL and AZ require formal firearms training including range time. UT requires firearms familiarity using real firearms and dummy ammunition, but no range qualification. The UT permit is probably the easiest to get and cheapest to renew ($15). A google search will take you to a New York Times article about a PA citizen with a PA permit who also got a UT permit to expand his ability to carry while traveling. The UT permit is often described as the most desired by non-residents (or at least the most desired by the man interviewed by the New York Times).
There are two "right denied" states; Wisconsin and Illinois. VT does not issue permits, but any citizen of any state can carry concealed in VT. There are 10 "may issue" states. Thus, there are 37 "shall issue" states + VT. On November 1, 2011, Wisconsin will join the ranks of "shall issue" states, leaving Illinois the only remaining "right denied" state. Please note that Alabama is "may issue" but most counties behave as "shall issue". Thus, on November 1, 2011, there will effectively be 1 right denied state, 9 true may issue states, and 39 shall issue states (if you count AL in the "shall issue" category) + VT which does not issue or require permits to carry a concealed firearm. The States of Vermont, Alaska, and Arizona do not require a concealed firearm permit for any citizen of any state to carry a concealed firearm. Wyoming will allow residents of Wyoming to carry a concealed firearm without a permit, but non-residents must have a permit that is honored by Wyoming. (Many such non-residents choose the UT permit.)
I hope this information is useful to you. (And thanks for the compliment with respect to the "Yellow Cat With The M-4".)