What are the pros and cons of cooking with a cast iron skillet?

The cast iron skillet is really great if you do cooking with heavier foods; those rich in fat or thick cut meats.

Here’s my general take of the positives:
Hot Cooking Surface
Yeah, I didn’t say even. I said hot. Look it matters. I have only ever had low to medium grade stoves, all of them electric. If you’re just using stainless steel, it doesn’t get all that hot. When you use cast iron, woo wee, it can get hot. That’s fantastic when I’m trying to sear a steak. None of my other equipment comes close to doing that as well.
Durable
I mostly have stainless steel but I also have some non-stick pans. I don’t use the non-stick pans anymore after I got the cast iron skillet. The cast iron is non-stick (once seasoned) but you don’t have to worry about scratching it and releasing torrents of carcinogens into your food by accident, and you don’t have to worry too much about other people using it because it’s really hard to damage it.
That being said, throwing extreme temperature differences into a cast iron pan is not a good idea.
Also, it can be in the oven. That’s useful if you’re doing forward searing of meat for a roast or a steak. I did that with a prime rib roast where I seared on the pan and then put the whole darn thing in the oven.
Non-stick
A cast iron skillet will let you discover what actually being non-stick is like. It’s way better than any non-stick pan I have ever used. Eggs just bounce off that surface.
Simple
Cast iron pans remind me of the simplicity of my stainless steel cookware. It’s just cast iron. Done. Whatever cast iron is or does, that’s what this pan does.
That said, it imparts iron into your food. Hard to say if that’s relevant to you.

Now onto the negatives I’ve experienced:
Hot Cooking Surface
You need to learn temperature control when using a cast iron pan. Since it can get very hot, you can also make much more horrible mistakes with your food. The margin of error will simply feel larger.
Cooking Time
As a large piece of solid cast iron, it takes a really long time for the skillet to heat up to the desired temperature. If you’re trying to get it really hot, expect to wait minutes. That’s sometimes quite painful in the kitchen if you’re trying to push out dinner faster. It’s a much more friendly tool during a weekend meal where time is not pressing.
Cleaning
If the person cleaning the cast iron skillet doesn’t know what they’re doing, watch them like a hawk. Like carbon steel, you cannot leave water on it and you must oil down the pan after cleaning.
So, first you scrub off any residue from cooking. You probably need to use water to get this done.
Next you boil off any of the water; I typically heat up the pan to boil it off.
Last, you reapply oil and then store the pan.
If you even submerge it in water while cleaning, you risk it rusting.
Weight
A lot of times I need to lift my cookware. For example, I might lift up a pot to pour remaining soup into a container. A fifteen inch cast iron pan weighs a shit ton. If you need to pour stuff it’s almost a two person job.
In that vein, I also don’t view it as a useful home defence weapon. Lifting that thing up to smack someone down will take forever. You’ll probably get shiv-ed twenty times before you laid a single blow. Yep, important information.

Here’s some stuff that I think you’ll hear a lot but honestly don’t matter.
Even cooking surface
Actually, most of my cooking equipment satisfies this but you’ll hear this pretty often. In most cases, an uneven cooking surface for cookware is really just shitty cookware. My stainless steel wok is an “even cook” surface but… so is my random small stock pot that I bought at a low-end grocery store.
Manufacturing Quality
It’s a stupid piece of cast iron. Any f-ing cast iron will do. In fact, all you want it to be is cast iron. That’s cheap ass shit. Unfortunately, in our day and age it might be really hard to find it for cheap because we’ve priced up worthless materials due to hype.

It’ll probably take you a month or two to get used to a cast iron skillet (assuming you cook at least 2–3 times a week). The cleaning is probably the most annoying thing and you’ll figure out how to do it fast when you’re forced to clean it nearly every time you cook.
The other thing to get used to is what temperature to set your stove at for your pan for various cooking temperatures. It typically gets a lot hotter than you expect and also takes longer than you expect to heat up.
I found it a positive addition to my cookware but it depends on what you value; I care a lot about the quality of my cooking but others care more about the time/maintenance.

10 Pcs Non Stick Cooking Set w/ Frying Pans & Saucepans