What's the best strategy for the Steam Monster Game?

The overall best strategy that all the top rooms use is simple, but it requires a very high degree of coordination, so unless you join an organized "pre-made" room, it will probably not be very effective.

1. At the start of the round, use all your badge points to buy two items ("Wormhole" and "Like New") in about a 5:1 ratio

2. On boss levels divisible by 100, stop doing damage entirely and just spam these items

3. Do not upgrade your auto-damage further than 10 Auto-fire Cannons (which unlocks the Boss Loot upgrade)

Wormholes skip a level when the current level is cleared, but if used on a level divisible by 100, they skip 10 levels instead. "Like New" clears all cooldowns in the lane, which can be used repeatedly to quickly dump more Wormholes.

By stalling these levels, you can queue up extremely large level skips to the tune of millions of levels, provided your room has enough Wormholes.

That's it. Upgrade-wise, your #1 priority should be health, because you can quickly get into levels where the monsters will one-shot you with millions of HP. Click damage and elemental upgrades are not important.

The Easy Strategy

I've been playing this game each day since the sale started, and I can safely say this is the best strategy:

Start a new game as soon as possible after 9am Pacific.

With this strategy, I've been to the highest levels I've ever been, exponentially higher than when I start a game later in the day.

For reference, in my current game, which I started around 9:10 am yesterday, I'm level 17,250 having clicked a grand total of 0 times. I have 50 or so of every consumable item, and have more money than I know what to do with. We clear levels so fast the spawners don't have time to show up on the screen before they're dead. Against bosses, we deal damage in the hundreds of quintillions.

The previous day I started around 11am Pacific and got to around level 250. I expect to shatter my record by a wide margin every day I manage to get a game started around 9am Pacific.

I suspect this works because the truly "hardcore" players start new games early, so I end up with a lot of people who have a lot of badge points and who have been at high levels in the past. (Like myself!) Having a lot of previously high-level players in your lobby means you can move very fast at first.

Stuff You Can Do Anyhow

Beyond that point, it matters a lot less what you do. My suggestion is to invest heavily into the "Light Armor" and "Auto-Fire Cannon" trees, as this will keep you alive longer and have you dealing more damage without having to invest much effort. I just get each skill to level 10 and then ignore that tier in favor of the next one.

Ideally here, you'd balance it so that you're only paying for the higher tier when it costs less than 10x the previous one, but again, way too much money for me to care. The first few ranks in any new tier of these skills are so cheap that it makes it very worth unlocking them.

Invest in elemental damage - ideally everyone would specialize so that "Max Elemental Damage" consumable has the most effect. I tend to peanut-butter it across because the team damage is so high it doesn't really matter.

There are a couple of abilities that have minimum requirements in the "Armor Piercing Round" tree, so I will grab ranks down there until I can get the Tactical Nuke and so forth.

With badge points, I used to stock up on Treasure items, as they help in the early game. Having a good dozen or so of these is helpful, for sure, but past that point it becomes annoying to use them up, not to mention the number you'll find from beating bosses. Past a few of each item, I tend to blow my badge points on expensive items just to use them up.

If you've got an insane amount of badge points, you might as well stock up on Wormholes - they can be used to very rapidly level up the whole game. I've got people in today's game who are using them so that every level we clear skips a good 5-10 levels easily. This might be suicidal if done too much in a low-level game, since the payout in return for level skipping is somewhat low. In a "power leveling" game started around 9am Pacific, it's not that big of an issue.

Use the badges you have gotten from the previous game to buy upgrades. Personally i would suggest to buy a mixture of the ones that give 100k gold and those that give 1% of targets money per click. I guess you can also buy a bunch of the Cripple Spawner items to get rid of spawners more quickly.
Use the 100k gold items to get yourself started. Use the 1% per click items on bosses (every 10th level). If you are going to check every so often, don't use up all your 1% items straight away... Save some for higher level bosses.

Do NOT use high damage abilities on bosses, since other people are going to use the 1% items to click themselves rich.

Which upgrades to get:
10x Light Armor
10x Heavy Armor
5x Energy Shield (this should give you more then enough health points even for high levels)
10x Auto-fire Cannon
10x Advanced Targeting
10x Farming Equipment
..x AFK Equipment (as many as you can get (10k damage per rank))
??x Boss Loot (own choice)
4x for each Elemental Damage
Don't focus on just the armor or just the auto-fire. upgrade them both approx at the same time. As you may have noticed, no click upgrades. Unless you want to click alot, it is better to upgrade your auto fire so you keep doing damage even though you are not in the game.

Abilities (if you are going to check on the game every so often):
Medics
Tactical Nuke
Cluster Bomb
Napalm

You will have to come back to the game every so often to check up on your money status and see if you can buy some more upgrades.
Also, check the prices for the different categories and go for the cheaper ones first (if you already upgraded to 4x elemental damage for each elemental type but not your auto-fire, you still don't do alot of damage.
If you got alot of money to spend ingame, do it on auto-fire / elemental damage options.