Going back to P mode because my pics suck!

I've been reading tons lately and have been trying to up my photo skills.  In doing so, I've left the auto mode and P mode behind and have been primarily using Av.  Unfortunately, my pictures are coming out terrible!  I am clearly not a natural when it comes to choosing proper settings.  In my defense, most of my recent shots have been in bad lighting situations - inside a restaurant and night time inside a home.  I'm running a T3i and using a Tamron 17-50 f2.8 lens, which I thought would handle the situation at a low ISO - 800.  It's not.  I'm having to crank up the ISO to 1600 or 3200 and then the shutter speed drops to 1/15 or lower.  When I review the photos and zoom in, most are coming out extremely grainy.  Also, the slow shutter is picking up movement everywhere and thus creating more blur.  I'm trying not to use the flash, but may reconsider.  In the end, I am thinking of moving back to P mode until I get a firmer grasp of my photography.  Any other suggestions?

adam