Yes, those will work great for you.

I started riding last year and rode all season using tennis shoes (a little stiffer than the running shoes I wear as my everyday shoes). No complaints at all. Then a few weeks ago a local shop had a good deal on some Serfas biking shoes (very stiff compared to my other shoes), so decided to give it a go. I even ordered these pedals to clip in to. While I don't regret doing it, I can't honestly say that I've noticed a big difference. It's still early in the season, so maybe once I'm back in good shape it'll be more significant. But the point it, your cross trainers will be more than fine for the type of riding you're doing.

Just get out and enjoy riding. Then upgrade clothes or components as you notice a need for them.