True, I do not have to upgrade the legionary, but once feudalism is discovered, the unit can no longer be built. It's replaced by medieval infantry in the production box...which I find to greatly reduce the legionary's value (it can only be used for less than one age!)...very disappointing.
The Medieval Age has 2 basic tech routes, the upper and the lower one. The upper one goes like Monotheism > Theology > Education > Astronomy and further. You can research pretty far up this tech tree while avoiding Feudalism. Feudalism is pretty useless for Rome, I can understand you want to avoid it. The only problem is that Feudalism is a prerequisite for Chivalry, and you don't want to be fighting knights while having none yourself. Well, personally I would probably want to be fighting with knights rather than legionaries anyway.
In terms of competing with the AI, what is the best way for Rome to keep up in the tech race, culture, and production at regent level and beyond, while at the same time keeping balancing with financing the military (which is probably the only key to winning with them)? How bout government? Personally, I find republic to be a worthless government in terms of warmongering, especially with its 3 gpt maintenance cost (I always plunge into immediate debt with republic). For Rome, perhaps the government line should go despotism, monarchy, then fascism?
Rome doesn't really stand out in anything particular, so it's up to you what you want to do with them. Your problems with Republic will be general empire management problems, not Roman problems. For Republic, build workers and road, road, road. Have you got 10 cities? Then you'd like 20 workers! Republic adds 1 gold to every worked tile that at least already makes 1, which is good money, but you need your tiles roaded.
This is something that most casual players have problems with. Also with creating enough settlers and workers in the early game. You'll need settler factories and worker factories to do this effectively. This involves graneries and some maths. If you don't like to get down to the nitty gritty of that, you can still play fine on Monarch and perhaps Emperor, with a bit of luck you can even wing it in a Demigod game, but getting the maths right does become more and more important as you move up level.
If you want to improve your gaming skills, there are several things you can do. There are articles in the war academy about empire management that can help. Also, you can upload a game save here, and you'll get more specific comments. You can also join in playing a succession game, where you play in a team, and you will learn a lot from the discussions in the team. In the succession game threads is a lot more discussion on strategy than in this part of the forum.