In 1.0 and later, it's largely through doubling upgrades.
The base rate is 0.5 CpS, or 0.8 after you buy Forwards from grandma.
This increases by 0.02 for each grandma after buying One mind, an additional 0.02 for each grandma after buying Communal brainsweep, and 0.05 for each portal after buying Elder Pact.
The rest of the grandma upgrades make grandmas 2^18 = 262,144 times as efficient.
- Six upgrades that double the rate are unlocked after having hired specific numbers of grandmas: 1, 5, 10, 50, 100, and 200.
- The nine grandma costumes are unlocked for having at least 15 of each building (Farm through Prism). These are also doubling upgrades.
- Buying seven grandma costumes unlocks Bingo center (quadruple); waiting after that unlocks Ritual rolling pins (double).
- During Christmas, the "festive hat" upgrade chain will eventually unlock Naughty list (double).
Once you enter the Grandmapocalypse by buying One mind, part of the grandmas' production actually becomes quadratic (O(n^2)) because the base rate for each grandma includes something multiplied by the number of grandmas, and that in turn is multiplied by the number of grandmas. With more than 40 grandmas, the majority of your production is from quadratic effects.
As with every other source of cookie production, this is multiplied by your cookie production multiplier (mostly flavored cookies and heavenly chips), egg multiplier (Easter-related), covenant multiplier (95% if in effect), frenzy multiplier (usually 100%, occasionally 700%, 50%, or 66600%), and wrinkler multiplier (100% to 600%).
Sources: "Grandma", "Upgrade", and "CpS" pages on Cookie Clicker Wiki
Without any upgrades or anything, a Golden Cookie spawns m minutes after the last one disappeared, where m is a random number between 5 and 15. Each of the upgrades Lucky day and Serendipity cut that time in half, and the debug upgrade Gold hoard makes them spawn pretty much constantly instead. Also, if your last cookie gave you a cookie chain, then the next one spawns 3 seconds later. This means that you are not more likely to get a Golden Cookie if you haven't clicked on one in a while.
Without any upgrades, again, a Golden Cookie lasts for 13 seconds before disappearing. The Lucky day and Serendipity upgrades both double that. If your last cookie gave you a cookie chain, then it disappears after 6 seconds instead.
When you get what appears to be a flat number of cookies, it either gives you 15% of your banked cookies or the equivalent of 20 minutes of cookie production, whichever is lower.
The bonus you get is randomly chosen each time you click the cookie. It normally chooses between Frenzy, which multiplies your production by 7, and Lucky, which gives you cookies as I just described. There is also a small chance of getting a Click frenzy, which multiplies your cookies per click by 777, and an even smaller chance of getting a Cookie chain, which gives you 6 the first time you get it, 66 the second time, etc. until you get something else. As Michael Madsen's answer points out, the effects appear to alternate because every time you click a golden cookie, there's an 80% chance that it will remove the previous effect from the list of possible effects before it picks which one actually happens.
Once the Grandmapocalypse begins, you also have a chance of getting Ruin, which makes you lose half as many cookies as you would gain from a Lucky cookie, Clot, which cuts your production in half, and Elder Frenzy, which multiplies your production by 666 for a few seconds.
Best Answer
My extremely basic test involving 25 clicks resulted in an answer you won't like, resetting does wipe the stat "cookies clicked". The only thing that stays is cookies baked (all time), and "cookies forfeited" goes up with the amount of cookies you gave up. So, just keep clicking without resetting.