Hi Duchess,
The Children's Museum is great fun, but a 14yo would maybe find it a bit juvenile.
The Indy Zoo is a good one, especially the dolphin aquarium, where you can walk under and see them swimming overhead.
The Indiana State Museum is nice, and has an Imax theatre inside.
The Indianapolis Museum of Art is OK - not as good as Cincinnati, but more than you would expect from sleepy-town, as we call it here.
Shopping-wise, there is Circle Centre downtown, with lots of specialty shops and chain food.
Restaurant-wise, we are the town with one of the highest concentrations of chain restaurants. You name, it we got it.
Aside from chains, we have several good locals:
Bazbeaux Pizza (with DwnTwn/MassAve + Broadripple locations) for thin crust heaven. $$
Scholar's Inn (also DwnTwn/MassAve + Broadripple locations) for high end food. $$$
The Rathskeller (DwnTwn/MassAve) for good quality German food, and a massive array of import beer. $$$
The Slippery Noodle (DwnTwn), for ambient jazz/blues (not sure about the food)
St. Elmo's Steakhouse (DwnTwn) for world class steak, and the best (spicy) shrimp cocktail you will ever taste. $$$$
Cinema Grill (86th and Ditch), for a unique Dinner and a Movie experience, all at once.
Cafe Nora (86th St @ Nora Plaza) for a little bit of French, a little bit of Italian. Great food, but $$$.
Mug-n-Bun (West Washington St), for old school burgers, footlongs, and root beer, served lke it should be -- car-side. $
Union Jack (Broadripple and Speedway locations) for an array of American favorites, + really, really good deep dish pizza. $$
Thai House (NW Side @ 86th & Michigan Rd) - the best Thai food Indy has to offer, but little ambience. $
El Rodeo (multiple locations) - good Mexican food. $
Squealers (NW side, 86th & Moller Rd) - award-winning barbeque. Try the spicy barbeque nachos. Trust me. $$
I could go on, but this is making me hungry.
DW