Those modes won't use flash unless you explicitly tell them too. Because you're no longer in a fully-automatic mode, there's leeway for the camera to adjust other exposure settings to (try to) let in enough light / make the picture brighter.
If you want the flash to fire, press the Flash button before shooting and the flash should pop-up. Its #3 here: http://www.imaging-resource.com/PRODS/T3/T3A2.HTM
The problem is when I press the button, the flash will not pop-up. When I press the button, the ISO setting screen appears. I have tried every setting I can think of. I have to go to automatic mode and push the picturentaking button halfway for it to pop-up
Somehow one of your "Custom Functions" got set (from the previous owner, I'd assume). From Page 77 of the user manual:
More details from Page 222:
Just hop into the menu, and change this back (to "0: Raise built-in flash"), and you should be good to go!
I have a similar issue with the flash not popping up under any circumstances. I checked Flash Control in the menu and it appears to me that a previous user (my daughter) may have used an external flash and set up controls, and then didn't cancel them out - because on all of the External Flash function menu items, I get an error message saying "this menu cannot be displayed. Incompatible flash or flash's power has been turned off." Any way to reset or trick the camera to think there's no external flash so that, hopefully, the pop-up flash is there when I might need it?
Reset the camera. The number of setting you'll have to re-enter is minimal. If the camera still believes there is an external flash connected, that's another story. The hotshoe may be damaged or require service.
Assuming you have a T3, See page 164 and 216:
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"Reset the camera.''
This is the correct answer. The fact you didn't know the flash setting was wrong indicates you may have something else incorrectly set. Reset will put you back to square zero. Menu>Tools>Clear all settings and also do a Clear all custom settings.