If I purchase a Google hub and set up everything I need to use Google Home to turn my house into a smart house. Will Google home turn on my smart TV when I issue the command?
This question is not a clear yes or no. Even if you have a smart TV, it may not have any features that allow it to be turned on via the internet. Even some Android TV models that have a chrome cast built-in do not always work with Google home.
You will need to look up the TV brand and model that you are using. The manual should specify whether it is possible or not to use the device with Google home.
If you are unable to find some answers, a quick method to find what you are looking for is to check the Google Home app. When you go to add a new device, check to see if there is anything in the app list for your TV and if there is, you should download this app and see if it allows you to link with your TV.
If you have an LG TV, for example, there is an app called LG ThinQ. If you download this and connect it to the app, you will find that you are able to then link Google Home with your LG account. This will allow you to turn the TV on using Google Home. You will also find that not all LG TVs work with this app. So the answer to your question is definitely a mixed one. Just because you have a smart TV, does not mean that it is possible to link the TV up with Google Home.
If you are determined to get this working, a Chrome cast may be a cheap way for you to achieve what you are looking to do without having to purchase a new TV.