Can you wear wrestling/boxing shoes during class? If so, those would be a better choice than sneakers IMO.
"Learning" technique at home is a bad idea in most cases (unless you are very, very good at visual learning, or have a very solid base to work from). You need to have someone qualified actually teach you the techniques to begin with. Once you know them, then practicing/mastering them on your own/at home isn't a bad idea though.
Also, be careful where you are getting your information about technique. There is a lot of bad stuff on youtube (some really good stuff though). Check out this thread for some good instructional vids:
Finally, you need to get some drills to teach you strategy. You can have great technique (which you definitely should develop), but if you don't know how to apply those techniques effectively, they won't be of nearly as much help. Also, quality is better than quantity IMO. Someone with 1 or 2 really solid techniques and a good understanding of strategy/how to use them effectively will easily beat someone with tons of not so great techniques but no understanding of how to effectively utilize them.