This is probably obvious, but so important that I feel it's worth saying: do not connect it to the internet! The last thing you want right now is up-to-date software, and chances are very good that if it goes online it will update its software and install all the latest security patches.
You will need to connect it to a network in order to scan it for vulnerabilities, but make that a network that has no internet access.
I would start by doing some searching on the exact make and model, especially searching through the CVE database to see what may be out there. There. If your TV has been connected to the internet, it may have had its software updated to patch any cves, but if it has not been connected, then there's a good chance it is still vulnerable and you can exploit them to get root or further access.
You can also throw scans at it. I would start with nmap and scan all the ports, also do service recognition to try and figure out what exact service is running on the other side of the port. For something like a TV, I would not expect a high success rate with identifying, but it's easy to run. What you can identify, the most important part is typically the version number. You can take that version number and compare it with CVEs with a lot more precision to see if there are vulnerabilities.
You can also try any number of scanners on it, such as nessus or openvas. There are tons of scanners out there so it's definitely worth doing some searches. I would suggest looking at the Kali Linux list of scanning tools, and either running Kali on a machine you have laying around, or use it with docker. If one of these scanners actually crashes the TV, that is ironically a great sign for your purposes.
If the TV has been connected to the internet, and you aren't able to find any vulnerabilities, it might be worth keeping it off the internet for a while to give some time for new vulnerabilities to pop up. That does require a long-term commitment to this project, but it's not like you can't use the TV. I don't connect mine to the internet ever anyway, because I don't want it spying on me and I hate its ads and crappy built-in software. I just use it with a Chromecast with Google TV and good old HDMI.
Depending on what you want to do, it's also worth going thoroughly through the menu and looking for any sort of developer or debug options. Sometimes these menus are very hidden, requiring on occasion weird keyboard incantations in order to even appear as options, but once you get these enabled you can connect using tools like ADB or SSH, and get a shell on the machine.
All in all, good luck! It sounds like a very fun project. It's a shame we don't live near each other because this sounds like a fun weekend project :-)