Active Playthrough (Clicking) Strategy
This strategy is intended for the active player, although it can also be used to increase general idle damage. The core of this strategy is to increase the amount of damage gained by actively clicking the wall, and how to spend your resources in the most effective way to boost overall click damage.
The game starts
Ignore all upgrades except for punchers (this will be the main form of damage and cash generation throughout the game). Increase Punchers to level 50, buying all upgrades along the way, then proceed to increase all other buildings to level 50, one at a time, so as to unlock each one's level 50 upgrade. (The level 50 upgrade affects hand punch damage by a percentage equal to the amount of each particular building owned. I.E. 50 clubbers = +50% to hand damage, so on and so forth.) Once this has been accomplished, buy all other upgrades to increase general idle damage.
When purchasing buildings, only do so with the intent of unlocking the next tier upgrade. Do not buy the cheapest upgrade available solely to increase damage immediately; you will end up spending more resources doing so (for example, clubber level 51 is cheaper than puncher level 98, but it will cost you more in the long run to upgrade clubber now instead of reaching the puncher upgrade at level 100 and then buying clubber level 51; $5 for a clubber level is cheaper than $45 for a puncher level, but $8,900 until the next clubber upgrade is more expensive than $3,600 for the next puncher upgrade).
Each low-tier building (up to Titans) you buy increases your hand damage by 1%. So, once there are no more easily reachable upgrade thresholds in sight, you can spend your extra cash on whichever building is cheapest, solely to increase clicking damage. Buildings after Titans only increase hand damage by 0.5%, so only buy those to increase idle damage.
An exception is the Brick Factory. Brick production is important for buying Necromancers, Demons, and some of the mid-game and late-game upgrades. Don't neglect brick production entirely.
Idle Damage Strategy
Idle damage alone will never be as effective as active clicking damage, by design. Nevertheless, for those times when you can't click the wall, it's nice to let the game play itself for a bit.
The first focus here is buying the highest tier buildings you can afford, as you almost always get more damage from cash spent on higher-tier buildings (in particular Black Obliterators) than from the same amount of cash spent on earlier buildings.
Upgrades are hugely powerful in this game, so you want to buy all of the DpS upgrades you can. This is especially true of the Synergy Upgrades, where DpS from one type of building is multiplied by the number of another type of building owned.
Due to how the synergy upgrades work, you may often find it more effective (higher DpS increase) to buy a Reality Compromiser which increases the power of each Black Obliterator you own, than to buy another Black Obliterator. Likewise, buying Brick Factories will increase the output of each Demon you own, and your Necromancers and Titans both boost each other quite nicely.
Rewinding Time lets you unlock new damage-multiplying upgrades (trophies and time bear wards). Since you keep most of your upgrades, you can surge through the early walls very quickly. It takes a few hours to get caught up most of the way to your pre-reset building levels, albeit hopefully with somewhat higher DpS.
Before you reset, try to purchase any unlocked upgrades (since all your cash is going away anyway). Sell buildings if you need to, because those are also going away. There is a strange bug that prevents you from selling Black Obliterators if the buildings scrollbar is all the way down (the mouse pointer flickers); try sliding the scrollbar up a centimeter or so.
Try to wait until you have at least 50 Cosmic Knowledge before doing your first reset. This gets you the first 4 trophy upgrades. If you want to hold out longer, that's OK too.
After that, you can judge when you want to do your next reset either by additional walls destroyed (one recommendation is to reset when you've gone 3 walls farther than last time), or by Future Knowledge breakpoints (waiting until you have enough capsules for the next trophy). Some players reset frequently, while others do it rarely.
Once you reach the Finality walls, gameplay changes in several ways. Some of the most powerful upgrades only become available at this stage; a few even require that you have defeated the Boss Finality wall (the 50th and final wall) one or more times.
After beating the first Finality wall, you will start to unlock "random points", and the upgrades to make use of them. Your random points are lost at each reset, and do not begin to accumulate again until you've re-destroyed the first Finality wall. Because of how powerful the random point upgrades are when you have lots of points, there will be an incentive to leave the game idling for long periods of time. The final random point upgrade (for the dream buildings) does not become available until you've defeated the Boss Finality wall twice.
When the walls are all destroyed, you will no longer have a use for Damage, until you buy Prestige Hacker (which requires beating the Boss Finality wall twice). This lets you convert Damage into cosmic knowledge. In addition to unlocking the final trophies, cosmic knowledge continues its importance by powering the Tellurium Trophy ("Everything is 0.005% more efficient per 1,000 future knowledge capsules").
Prestige Hacker channelling damage into cosmic knowledge, which in turn increases damage, is the final tool for reaching the real end of the game: infinite resources. At that point, you have transcended the game to the point where it is no longer playable. Congratulations.