How to I know if Wifi is enabled on my Curve? Is there an icon on the screen like the bluetooth icon?

You would see a WiFi symbol in the upper right near the wireless signal.

Or, you can check under "manage connections"

Thanks.

This is going to sound really stupid, but here goes. Since I don't have 3G, if I leave the wifi on, will the browsing speed increase? And if I leave it on, will it drain my battery faster?

No it will not do anything unless your are connected to WIFI.

When I am connected to wifi, will it drain the battery?

Wirelessly posted (Tmo's cheapest BB customer)

Assuming you are connected to a network/hotspot, wifi browsing does tend to be faster than normal EDGE.

As for leaving wifi on, I do think it will drain your battery quicker though it depends on the strength of your connection. Plus, if you leave it on but are not connected to anything it'll probably drain more just because its actively searching for a nearby connection.