When you are traveling, there is 99% chance you are going to connect to the internet via some else's WiFi - such as hotel, restaurant, library, wherever you happen to be. Hundreds of millions of people connect to these public type WiFi hotspots every day. The biggest step you can take for security is to have file sharing and network discovery turned off when connected to these hotspots. Windows is great because it has public and private network settings and you can turn off file sharing and network discovery on public networks and leave them turned on for your home network.
Whether you connect the Chromebook via Ethernet cable to your router, or just use WiFi is largely a moot point because chances are you are going to have to connect to the public WiFi hotspot anyway. Now, there some advantages to using a travel router - and if you use one, make sure it has WiFi bridging - meaning it will connect to the provider's WiFi hotspot and then repeat it to your LAN side, either on it's WiFi or via Ethernet. I have used this one in the past:
Hootoo HT-TM02 Tripmate Nano
The advantages are:
1. You can maintain network sharing on your LAN side just like at home. This is good for long trips with kids or other adults that want entertainment. It's USB powered so it can run off a computer or USB charger. You can plug a USB hard drive or flash drive into it and stream movies and such to WiFi devices like tablets or the Chromebook. I did this in our car during a long driving trip. I also did some trickery to share a cell phone's data connection through the travel router so anyone in the car could connect to the internet.
2. When at the hotel, it will connect to the hotel's WiFi, usually. This gives you internet capability on your LAN side and still maintains your LAN network streaming capability. That also puts another NAT firewall between your LAN side and the hotel's WiFi.
3. The travel router is better at connecting to weak WiFi signals. We were staying in cabins and the resort's WiFi was in the main lodge. The signal in the cabin was not strong enough to connect our computers to. I set up the router with the lodge's SSID and WiFi password, stuck it in a window, and went to the hot springs. When I came back, the router had managed to pull a connection from the lodge's WiFi and gave me internet in my cabin.
The downsides:
It takes time to set up. Connecting the router to the public WiFi hotspot is not easy because you first have to connect to the router to get into the setup webpage to enter their SSID and WiFi password and if they have the login webpage type security it doesn't always work without first connecting your computer via WiFi, gaining the connection, then spoofing your computer's MAC address with the router.
It is also another piece of gear you have to haul around with you and it does require power, either from the computer USB port or a charger.
On short overnight trips I don't bother with it but on our 3 week long vacation I was glad I had it.
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