The Vonage phone does not come with a WiFi adapter, but you can buy one separately.
So:
You need is a device called an Ethernet (LAN) Wireless Bridge, sometimes called an Ethernet Converter. Basically it is a wifi device that has an Ethernet port, so you can plug in something via a Wired Ethernet LAN cable. The Vonage devices use an ethernet cable to connect to the cable or DSL modem, so they can just plug into the Bridge and use a wifi network to access the internet instead.
All you have to do is set the bridge to connect to whatever wifi is available, then plug the Vonage device into the bridge with its LAN cable. Nothing to do or change for Vonage. Phone works just like a regular phone from 100 years ago after it is connected to the internet.
Buffalo Tech Airstation
Understanding Wireless LAN Bridges
There is also a way to use your computer's WiFi receiver to route the Vonage through, but that involves re-working your network settings, which I wouldn't recommend in your case. Just buy the adapter, but the information is on the web, if you prefer to do it. I have a friend that uses that exclusively and it works fine. But, there are very few hotels that don't have an ethernet plug-in or an adapter at the front desk for people who don't have WiFi capable computers.
As for the device and phone, you can pick them up at any Best Buy and most other computer stores... You can also purchase it directly from the Vonage site or through Amazon.com.
As for the extension to your mobile phone, it just requires that you have an existing Vonage residential account, then download the app to your smart phone. When you have a WiFi/3G/4G connection, just open the app and dial the number. Works like a regular cell phone at that point and the charges are charged to your Vonage account.
Highly recommend you browse
The Vonage Home Site to get an idea of how easy it is.
There are no tricks or gimmicks to it. It's really easy.
An alternative would be
The Magic Jack, which works very, very simply as well... Just plug into your USB port and plug the phone into it. I've used this, too and it works well, although I think Vonage is better.