Wow, new blood! Welcome
For the record both Shogun1 & Medieval1 are now available on Steam, patched to run on modern OS & cheap for that full retro experience. (personally I prefer the old-style campaign map with the simultaneous turns over the sequential type since Rome1)
Medieval2:Kingdoms is mainly of use for running mods like Europa Barbarorum2, Third Age etc.
I believe if you're running Arena OK then Rome II should also run fine.
Yes basic game should be a good start, probably I wouldn't buy it at the full price they're apparently charging atm, but then I've been playing since it came out... (Steam says 1400hrs)
The available factions is a bit thin in the basic game but does cover the main areas, a few notable absences but a few interesting quirky ones.
What factions are you particularly keen to play?
Aside from Grand Campaign the Emperor Edition has the Emperor Augustus Campaign which is a good intro to the Campaigns (major point is they are 4turns per year -> your characters stick around longer -> get ample time to rank up to max, tend to die off mid-tier in Grand Campaign)
The ingame info got a lot better last couple of patches but this site
https://www.honga.net/totalwar/rome2/ is still very good for exploring what units are available for a faction & what buildings you need to recruit them.
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