You do need an iPhone to update the Apple Watch, download software for the Apple Watch, etc.., as the Apple Watch is just an accessory to the iPhone.
The $10 per month fee is what the cellular carriers charge to share your iPhone phone number with your Apple Watch.
But if you want to leave your iPhone at home, you can go out with just the Apple Watch.
But you should be aware that using when the Apple Watch is near the iPhone, the Apple Watch conserves it tiny battery by depending on the iPhone's stronger radios, and bigger battery by using Low Energy Bluetooth to ask the iPhone to do the heavy lifting.
When you are away from the iPhone, the small Apple Watch battery has to power the more power hungry cellular radio. The battery gets drained even faster if you make lots of phone calls, texting, listening to music, etc... a lot when away from the iPhone.