shoes
To dance tango you need to be able to stride powerfully but also pivot easily on the ball of your foot. Traditionally that's done in flat shoes for the male leader and high heels for the female follower on wooden dance floors.
This semester we'll be in the dance studio, not on a wooden floor.
Socks should work fine. (Bare feet will likely be too sticky to get much pivot.)
If you would rather wear shoes, then you will need indoor-only dance shoes or dance sneakers, probably with a suede sole.
One choice would be these, the Sansha Salsette, which have worked for some students in the past.
They're about
$35 from Amazon. (The sizing is off though - order about 3 sizes larger than your regular shoe size.)