I've been carrying the torch around here for this topic. It can get a little lonely at times, since wide empires and Liberty are currently in vogue.
The short answer is that

in the capital is super important, especially early. The biggest mistake that many make (I used to do it too) is thinking that you build a

powerhouse through having lots of hills and building lots of mines from the get-go. Lots of hills are nice but not really required to have a good

capital, so long as you work enough tiles.
You must build a base of

early to get an optimal

capital. Granary should be your 2nd build (as soon as you get pottery, your first tech) in most starts. If you have 3-4 deer/wheat/banana tiles, this becomes the most important building you will build in your game. +5-6

on turn 15 is HUGE. You don't have to improve the tile for the bonus.
Go all in on

your first 50 turns. You are better off working 8 unimproved tiles than 4 improved ones (luxury benefits excluded). Hanging Gardens is your key wonder, which should come in somewhere in the 50-60 turn range. Fill up Tradition too, with focus on the

building policies. Don't worry about burning legalism on a monument - the

and growth can't wait. Nor can the awesome Monarchy policy, the best policy in the game for a super-capital builder.
After all this, you can run your capital in

focus for the rest of the game and still grow a massive population capital. Wonders like PT, HS, ND, etc. should be doable in around 10 turns each and medieval level buildings like Market, Workshop, University, Temple are all 3-5 turn builds when they come available.