So, I have my Bianchi San Jose completely apart for maintenance.....except I cannot remove the bottom bracket to replace it with a new one I got. I have the right BB tool/wrench thingy, and I know that it should loosen on the drive-side by going clock-wise, but it won't budge. I am wondering just how much torque it should take to unscrew it, because I don't want to destroy anything on my bike trying to do routine maintenance.

Is it common for BB's to get stuck? I did ride it in the winter here in MN.

They can get tight, as you've noticed. Apply a good penetrating oil like PB Blaster, Liquid Wrench etc. and let it soak a few hours/overnight.

a 4 foot breaker bar or pipe that will fit over your wrench works wonders. Barley have to push on it to break it loose. used htis method on many bikes with no problems

Once you remove it, as suggested by the other posts, use plenty of grease on the threads of the cups before installing it. Also, I often remove the bottom bracket twice a year to inspect, but mainly to exercise that joint and keep it operable.

Cartridge BB? Be careful with the tool/BB interface - if it's a hollow spindle, you can put a QR skewer through and hold the tool to the BB. For removal, it's often easiest to put the tool in a vise and turn the frame. As long as you're sure you're turning in the right direction, it's pretty hard to damage the frame. Same technique works well for removing the fixed cup from a cup and spindle BB.