Essentially, I am looking to make the right choice between Rome, Medieval and Shogun. I will describe my situation and invite your good advice. If you have suggestions outside of Total War, I will love to hear those too.
I am the only person in my group with any Total War experience. All of us have RTS experience, most of us (but not all) have 4X experience.
My group was heavy into wargaming boardgames, but in February I moved away from them and the group no longer meets regularly. We are looking for a wargame that we can play online. It would be fair to say that if Axis and Allies or any of the MB games had a good online version, I would not be posting here today. We would be playing those. I say this just to give you a feel of our general 'depth of strategy' level.
Because some of the group are poor, we are currently only considering the first editions of Rome, Medieval and Shogun. $10 a pop makes it an easy sell. Of the first editions, Rome is the only one that I personally played... but I loved it and imagine it holds up quite well.