You've chosen a great hotel! I worked at the TI Wedding Chapel for two years. You'll love it there! I've heard mixed reviews about the new show, if you do it, I'd suggest purchasing the premium seats, or allow 90 minutes prior to showtime to get a good spot from the gangway.
Rental car is right at TI, as mentioned. Based upon what you have said, I'd rent a car. Driving to the dam is under an hour, and it's a great tour. As for going to Mt. Charleston, it is approximately one hour from the Strip. Take 15N to 95N to the exit marked Mt. Charleston. it is BEAUTIFUL, and I suggest you have lunch at the Lodge---breaktaking view from the patio seating. NOTE: The Lodge, not the Hotel which is lower on the mountain! Another alternative is a side trip to Red Rock Canyon. Red Rock is about 45 minutes from TI, straight North on Charleston (two exits North of TI on I15). It's worth the trip. Parking at the hotels is GREAT, relatively convenient and FREE!!! Feel free to Valet (it is a rental!), just remember to tip $1-2!
Vegas revue shows: The Stratosphere has a passable afternoon show that is "free" with the purchase of a cocktail. It's a decent show, and it uses a piece of an afternoon, not an entire evening. Spend time at the Freemont Street Experience in the evening if you can. It's free and lots of fun---it's "old Vegas". The buffet at Main Street or the cafe in Golden Nugget are good eating choices Downtown. Crazy Girls is a good show, as is La Cage (both at the Riv).
OPENED 1/1/04!!! We now have a 1/2 price same-day ticket kiosk in town! It's next to the MGM on the Strip. Might be worth a try---I haven't tried it and don't know anyone who has, but it is supposed to be like Tickets Tickets Tickets in NYC!
Have fun and GOOD LUCK!!!
Patti