In my opinion, since there are plenty of ways to end the game before turn 300 comes, earning fame is not so important to win. So I choose next culture as soon as I collect 7 era stars(except when valuable eq was not built yet). After entering the contemporary era, vassalize all the other empires or launch some rockets to the space, end the game before others have enough time to collect era stars.

Then, you are either advancing with a science rush and leaving almost everything else behind, or going a bit the war path and missing technologies. You'll always be at least behind players who do get both science star + military/expansion ones. And, by the way, many players who do not focus on advancing do easily beat the game before turn 200. Having 300 turn in mind is quite wrong if you don't play on endless speed.
May I ask how much fame you do get by the time you're finally winning?

I personally use a mod that allows multiple people to choose the same cultures in an era, and it has completely changed the way that I play this game. You no longer feel rushed to move into the next era because you might miss out on the culture you want. I will admit it does have it's own challenges when you see 4 Harrappans, 3 Myceans, and whatever else the AI picked in the first era.

But to answer the question. The way I play is I will usually go for all 21 stars in each era, or at least research every tech available before moving on. If you are switching when you get the minimum of 7 stars (which is super easy to do) then it just feels like rushing through the game and not enjoying and maximize everything you can do.

With this game play style I can have thousands more fame than any other AI, and fame is the only true win condition. You can set what your end conditions are, but it's fame that gives you the win.

Other Ai's could pull ahead in fame at the beginning, but it's easy to pass them eventually. Whenever I see an AI start to get a lot of points they become my new war target and take them down.

For me it really depends on where I'm at in the game, how accessible the stars are and if the next culture I want is really important to me.  If I REALLY want that next culture I'll go almost immediately.  I always double check that another star or two aren't 1-2 turns away and quickly gobble those up.  Otherwise my goal is to get as many as are in my plan for that era.  So I may not be money or science focused during an expansionary era where I'm at war a lot so I ignore those while I scoop up all the military and territory stars.  When I'm at peace I'll get all the others, and so on - I rarely if ever get ALL the stars - it would take too long and involve awkward hoop jumping strategies. 

Aha, I see, I just played another multiplayer game now, and I decided to check which stars are close to attain after I have reached 7, and I wait to get those extra ones, (around 3 or 4 more). But I made sure that I get my cultural stars which give more fame. In the last game I completely ignored that. Now I'm in the first position by the 3rd Era. the random guy with me had to leave, but it was a fun 4 hours.