China
Most hotel lobbies and rooms will have free wi-fi internet access. Some restaurants and coffee shops offer it as well. Getting access via your U.S. mobile provider is generally very expensive. The exception is T-Mobile which has free international data on some of its plans. Some may wish to purchase China SIM cards that include phone or data. This is pricey (about $50 for a week). This is done at the airport and requires your phone and passport. It's generally not needed nor recommended.

As you know, China limits what people within "the great firewall" have easy access to. This includes major social media services such as Facebook, Instagram and Twitter, virtually all Google services (search, email, maps, Google Drive), and many other news sources (e.g., NY Times, Bloomberg) and tools (e.g., Dropbox). Microsoft and Yahoo services (email, search, cloud drives) are generally not blocked.

In order to access otherwise blocked web sites or services, you need to have a VPN (virtual private network) that you turn on before trying to visit the site. It works by routing your search request to a server outside of China and then giving you information from that site through that server. Sometimes even these do not work.

If you plan to use any blocked service, you should download and install your VPN before leaving for China. If you wait until you are in China, you will not be able to find such apps in the Chinese Apple App Store and Google Play (for Android) is blocked. There are many Chinese app stores, but all are prohibited from offering VPN.

There are free VPN. There are paid VPN that offer a free trial (usually a week, which will cover most of our time in China) and you can subscribe to some VPN for just a month. Some VPN are more user friendly than others. I have used three different VPN in China: USC via Cisco AnyConnect; Astrill, and Witopia. All have worked for me and all have sometimes failed. I recommend downloading, installing and trying a VPN before the trip. It's safer to use a VPN even in the U.S. if you access the net via wi-fi at school, the library, coffee shop, or elsewhere. You need to make sure you know how to use it.

Most of you will rely on mobile devices. Most VPN also have desktop applications as well, should you choose to bring a laptop. You are strongly advised not to bring a laptop.

Taiwan
Again, most hotel lobbies and rooms have free wi-fi internet access. Every coffee shop and many restaurants offer it. The same situation applies in terms of using your U.S. carrier or getting a Taiwan SIM card.

Taiwan has no great firewall. You may still wish to use a VPN, however, in order to increase the security of your communication.

Of course the same advice applies in China and Taiwan as in the U.S. with regard to using your credit card, etc. online. Check always to see that the site is secure (your browser should show a lock and the web address should begin with https://).