The information on the cards is straightforward, with you "reading" them from top to bottom, left to right. In the image above, acquire an upgrade token, optionally sell oil for 4 money, optionally spend an upgrade token and coal to upgrade (i.e., flip over) a card in play, optionally sell two coal for 2 money up to two times, acquire two iron, etc.
How do you gain these cards? The first half of each round is an auction in which players take turns placing one of their four bidding tokens — conveniently numbered 1-4 — on one of the available cards, with you being able to bid on a card at most once and with players being unable to place the same number on a card as someone else.
If you place the highest number on a card, you acquire it and can use it once during each of the subsequent production phases; if you bid on a card but didn't acquire it, you use the compensation effect at the top of the card as many times as you bid, so (reading from the top line, left to right) the red player receives one iron, the yellow player two coal, the black player two coal, and finally the red player can convert iron to oil twice while the yellow player can do that up to three times. Cards are placed in order for the auction, so the red player can use the iron acquired on card #2 for the processing on card #7.
The auction in Furnace is brilliantly minimalistic, with each bid mattering and with you having advantages no matter your position in turn order. You can lock in a card as yours by placing your 4 on it — but then the next player knows they can place their 3 on it to get the compensation three times, and late in the game compensation (usually) trumps acquisition since you'll produce with that card only once. That said, if you don't place the 4, someone else might grab the card, which could be the only oil-processing card available that round, so maybe you have to give to get...
You're checking out each person's engine and resources on hand to guess what they want and how they're going to bid and how you can take advantage of that — assuming they do what you'd do if you were them. Sometimes the cards on display force you in a particular direction; if only one processing power for money turns up, then you need to pump out the raw materials to prepare for the next round, ideally shifting up the raw material scale to oil, which returns the most money per unit.
Notice in the image above that the bottom line on each card is grayed out. When cards are placed in auction, you have the compensation effect listed at the top and the basic effect immediately below the image. The greyed-out line shows what you'll get if you upgrade the card, and the image at top shows upgraded cards.
Your goal in Furnace is money, so don't think you win by acquiring cards, which is evident in the image above as my opponent added only five cards to his production chain compared to my eleven and his score was one-third larger than mine. (With two players, you use a die to simulate the bids of a dummy third player — a system that works far better than I thought it would. I have an extra 2 bidding token thanks to my unique capitalist power, whereas my opponent can spend coal to produce twice with one of his cards.)
Furnace is all about efficiency, with you not wanting to leave any resources behind where they will count for nothing more than a tie-breaker — and in the second half of each round, you can re-arrange your cards as you wish to co-ordinate the inputs and maximize the outputs. As such, this production phase is the mirror image of the auction phase: no interaction, just head's down calculation to see which arrangement of cards produces the best results. Note in the image above how we've arranged the raw materials above each card to figure out exactly what goes where. You might even forget that others are at the same table...
For more on the capitalist powers, the two-player variant, the dichotomy between the two halves of a round, and the variant that somewhat lessens that dichotomy, check out my video overview of Furnace, which has been released in Russia by original publisher Hobby World and which has been licensed by multiple publishers around the world, with a U.S. release from Arcane Wonders coming in Q2/Q3 2021.