Not free yet but will grab it when it is.
As a new PC gamer, what are the other Total War games I should check out?
The Warhammer games for the battles - it's got huge unit variety and is by far the most in depth, however the campaign map gameplay is hit or miss. Lots of DLC factions to pick from later on, but don't worry about getting all of them. Pick out ones you like on sale, and it shouldn't be too expensive. You can also play a combined map of both titles along with all factions/dlc factions (and the later 3rd title when that's released) if you own both games.
Three Kingdoms for campaign - This can also be hit or miss in the campaign department, but is currently the best in that aspect as far as I'm concerned. There's a lot of room to build a narative with your faction here, I would say diplomacy is actually pretty fun in this case, and battles are great too. Battles though lack variety, and that's a major sticking point for Warhammer players, because it can legit feel like a big step down.
Then you can go into whatever period you like I reckon. Here's just a further breakdown which might help:
Fall of the Samurai - standalone dlc for Shogun 2 - will be a comfortable change for you going from the base game, with guns and gunships being a big change of pace for the battles. Very Gun focused.
Empire - Again, has guns, but you can play more with army composition than Fall of the Samurai. Campaign map spans most of the world, so things can get very expansive.
Napoleon - More focused Empire centered on Europe, with some improvements gameplay wise.
Rome 2 - Europe focused, with some of North Africa and West Asia in the mix. This game has some in-depth faction mechanics that warrant a special mention. Lots of unit variety here. DLCs are generally quite good with some unique mechanics added in some cases.
Attila - Like Napoleon and FotS, but for Rome 2, this focuses on the decay of Rome, and horse archer spam! I personally really like this game, it's growth/food/climate mechanics, but it's an incredibly flawed experience that has very poor performance. I wish it got the same post launch attention Rome 2 did. DLCs are decent and have some new mechanics - one centered on kingdoms called Age of Charlamagne, one called The Last Roman Campaign, which is a very narative focused about the Roman expedition to retake Rome from the Goths. Avoid the faction DLCs however, they suck.
Medieval 2 - Europe focused but this time it's medieval! This is also the point where you're really going to notice the age, but this is also the game with the biggest modding community, which is why it's still recommended.
Thrones of Brittania - Very much an experimental game for Three Kingdoms that you will wear thin very quickly. Don't recommend this.
I'm no authority on this, but that's the best explaination I can give without really getting deep into each.
Also just a side note, none of the DLC for
this game are really worth it outside of Fall of the Samurai, unless you really want a new starting position or, in Rise of the Samurai's case, a slightly different campaign that's very similar to the base game.