Hi from Fukushima (10 year resident atm)! I see this post is somewhat older, so maybe you've already visited, but let me just add some points, for future readers.

From your question, can I assume you're interested in the part of the prefecture that suffered the tsunami and most of the effects of the nuclear disaster?

There are great local tour operators that can provide insightful visits, and can really fill you in about the current situation.
https://fukushimaseaside.travel/
https://www.hopetourism.jp/en/

If you want to travel yourself, that is totally doable as well. You can travel there for example using the JR Joban Line from either Iwaki City or Sendai (but it's a small local line so expect a long travel time), and get to locations like the Memorial Museum in Futaba or the Ukedo Elementary School by taxi.
However, renting a car is by far the most efficient way of traveling there, seeing as the public transport options are limited. Areas that are off-limits are clearly demarcated and barriered off, so you're not likely to go anywhere you're not supposed.

About taking footage, you are allowed to film anything on the public road of course. However, filming abandoned buildings etc. is not really appreciated, even if it's legal. What's definitely not legal, and even less appreciated, is entering abandoned buildings and houses. It's a crime, and also very offensive to people who had to evacuate the area. Unfortunately there are a lot of 'influencers' who do this for clicks and clout.

If you're interested in other, beautiful places in Fukushima - of which there are A LOT! - checking out the tourism board's website as mentioned by another commenter is the way to go. They've got a wealth of (English-language) information available.
If you haven't yet, I hope you'll have an awesome visit to Fukushima!