To start, I've run track and xc in high school and now going to be running at a D1 college.
I suggest getting a good pair of training shoes from trusted brands such as Saucony (my personal choice), Nike (like the kinds you are thinking about), Asics, Mizuno, or New Balance. The perfect shoe for you depends on how you run (your stride) and what feels the most comfortable for you (some people need more support others don't). Is there a running store near you that you can go try on and run around in some different brands? Try a bunch and see what you like the most.
I also recommend getting a pair of spikes for racing because they allow you to dig into the ground getting more of a push and maintaining control in the cross country terrain.
But I can tell you, running in tennis shoes is a bad idea and you need to get legit running shoes asap!
Rs Emperor