Where should I install sagemath?

Sage's installation instructions basically tell me to just untar it and run it from wherever I'd like. Not being experienced with the Linux way of where things should go in the filesystem, I'm kind of at a loss where I should best put it. Putting it somewhere in my home directory feels wrong.

Where would you extract it to? /opt?

The Linux Filesystem Hierarchy Standard that is part of the Linux Standard Base recommends to use /opt. I have only had good experience with that. Many commercial packages go into opt as well and stay to their own folder in there.

/opt is a viable option. Some people install it under /usr/local/

I use SAGEMATH too. Actually, you can put it anywhere you like. I usually just put it in my home foler. Exp, '~/Softwares/SAGE'. And of course, you can make some links for convinence.