Nice tip on the 2n+1 rule Mitchum!
It's not my rule, but I'll take credit for it anyway...

It doesn't always work, but it gives you a pretty good chance since one unit can soften up each defender and the second one can kill it. Of course, if you don't scratch a defending unit on your first attack, it's time to panic... after you pick your keyboard up off the floor and retrieve your mouse from across the room, of course!
By the way, if the AI has units near the city that could possibly capture the city back on the following turn, I may kill all but one severely wounded unit on the first turn and then take the city the next turn so that I can move several units into the city to heal and to ensure that I don’t have to capture it again.
Do you find it's generally best to go for the enemy capital first rther than the smaller cities?
I don't have a set rule for this. Sure, if you can take an AI capital and break his back, do it. However, I typically take the city that is closest to me and work my way through his cities in what makes the most sense with respect to unit logistics. Note that if do you take the capital first, another city will become the capital, which just could pop the borders of said city before you can attack it, making it that much harder to do so when the time comes.
On pillaging, my preference is to just pillage the strategic resources, so I don't have to redevelop everything over once I get the city. What's your usual practice?
If I'm planning to keep the city and intend to capture it soon, I don't pillage anything since I want to keep all of the improvements without having to use any worker actions. If I don't have a 2 movement unit, I typically don't pillage anything either since it slows down my stack. If I'm just fighting through an AI's territory to get to another AI on the far side of him, I'll pillage him back to the Stone Age and come back and take the cities later once I have enough units (usually way more advanced the next time – i.e. macemen and cats vs. archers rather than axes and swords vs. archers) available to do so. Lastly, if I'm hurting for gold because I've kept too many cities or I'm fielding a very large army and my expenses are high, I'll pillage cottages for the extra gold.
And on ways to reduce the play-time of the game: I noticed in your log that units were promoting after battles automatically. Do you generally use the auto-promote feature? What a "Stack Attack", do you use that? Any other tips to reduce the number of clicks it takes to execute a Conquest game?
I typically save all promotions until right before I’m going to attack. What you saw in the log was me promoting the unit right before I planned to attack, not after. This does at least 2 things. First, it lets you take the promotion that will provide the most benefit. If you’re facing an archer, you can give your unit a Cover promotion. If it’s a melee unit, Shock works. Of course, City Raider is always good no matter what unit is defending as long as you’re attacking cities and not units out in the open.
If my odds of success are very good (typically better than 90% or so), I won’t even use a promotion at all. That way, on the following turn, I can use the promotion to speed up the healing process of said unit such that he can join the battle that much sooner.
I have never used the auto promote or stack attack features (unless you count mis-clicks

) and don’t recommend doing so. I prefer to have fine control over which unit attacks and which promotions to take depending on which unit is defending.
I don’t think I play all that fast and I would actually prefer to slow down most of the time to avoid the silly mistakes that always seem to find their way into my games. Maybe someone else on the Forum can provide you tips on how to play faster or you can just go
here, courtesy of The Me In Team.
Although, that begs the question: Why do you want to play
faster? If you want to play
better, slow down and enjoy the process. Check every city more often to make sure your citizens are working the proper tiles. Focus on your worker management. Take notes. Compare your game to others (I think you've got this part down!). Learn from the community. Play the game again to see if you can improve. And most importantly, have fun!