I've watched some videos. I've read some articles. But I still don't know how to figure this stuff out in real time while I'm at the table or playing online and possibly many tables. Is there a trick?

these are the basic ways that cards form poker hands. if each dot is a card, then the lines connecting the dots represent hand combinations for holdem.

Start with the basic formulas and then practice them off table. This article goes over it pretty well.

Unpaired hands: Multiply the number of available cards. (e.g. remaining combos of AK on an AT2 flop = [3 x 4] = 12 AK combinations).

Paired hands: Find the number of available cards. Take 1 away from that number, multiply those two numbers together and divide by 2. (e.g. TT on a AT2 flop = [3 x 2] / 2 = 3 TT combinations).

Edit: Bob's method is more intuitive and aesthetic. I like the graphical method.

The following has been extracted from the Combinations article in the Hold ‘em Mathology blog on Tumblr. It may be useful for those comfortable with high school math,

https://holdemmathology.tumblr.com/p...r-calculations

Combinations are the number of ways hands or boards can be dealt, without regard to order. Qc 8d for example can occur with the Q being the first or second card. But even though that is two possibilities, disregarding the order makes for just one combination.

The combinatorial formula for dealing r cards out of a total of n cards, or choosing r things out of a total of n things, is

C(n,r) = n!/(r! * (n-r)!), 0 =< r <= n, n > 0

where x! = x * (x-1)*(x-2)* … *3 * 2 * 1 ( e.g., 4! = 4 * 3 * 2 * 1 = 24)

Other notations for combos include nCr and n Choose r. Note that C(n,0) = 1 for any n.

Example: How many pairs are there for a particular rank. A rank has 4 cards. A pair is gotten from any two cards of that rank. So, n=4 and r=2. Therefore, the number of combos is 4!/(2!*2!) = (4*3*2*1)/(2*1 *2*1) = 6. Of course, for small n, combos can be gotten directly by counting. For a pair, the enumerated 6 combos are cd, ch, cs, dh, ds, hs.

How can I figure it out while at the table, just on the fly?

Like if I have KQo on a Qs9s3c flop and get check-raised or check-called, how can I think here in terms of poker combos to help me if I have less than 30 seconds to act?

Normally, you use combo analysis off line – researching, review of hands played, setting up baseline strategies, developing combo tables, etc. On-line or live, it is unlikely that you can do effective combo analysis. Not only do you have to count the combos but then you have to use this count in making a decision so it is at least a two step arithmetic process.

Yes, there are times you can do it such as when you just know that villain’s range is QQ+ and based on the board and your hand, you can count the winning and losing combos and do a quick ratio calc; but, such occasions are rare.

One simple approach sometimes possible is to count outs, the number of cards that pretty much assure a win, and use the 2x -4x rule; your equity is roughly twice the number of outs for a turn or river decision, or 4xOuts for a flop decision. It’s usefulness diminishes if the bet is not all-in or not a closing river decision.

This or just use your fingers.

I do both:

L(r,i) = Rr1k/(p2k * (m-i)k), Lp<Rp<Rik, t>m

L=Left
R=Right
r=Ring finger
i=Index finger
k=Knuckle
p=Pinky
m=Middle
t=Thumb

To be honest, my advice ist just practice, practice, practice.
It sounds cheesy but learning by doing is your best chance to get better. I'd suggest to buy a poker set an practice at home.
Here's a great German article, about different Poker Sets, that you can easily translate with Google Translater. Maybe it helps you with your decision.