Hi April Showers,
Welcome to the forum! You have a phalaenopsis, also known as a moth orchid. Here is a website:
For general orchid culture (from the FAQs)...
On the left is a green box labeled "Growing Tips", and click "Phalaenopsis." You can find specific tips regarding phals. Sounds like your phal is doing great! New leaves and new roots are always good signs. Phal roots tend to grow out of the pots and all over the place--that's all right, it actually means the phal is happy!
If you keep it happy, it'll bloom easily. Welcome to the orchid world! Phals are a great start. Just two important points to keep in mind: 1) don't overwater. 2) Give it enough light. The FAQs give very good info to let you know more. Good luck!
Good luck with your orchid!