TL;DR For euchre, memorize the suits in the same order every time and remember the boss cards in each suit plus all trump cards.
For counting in Euchre, which I think is pretty easy to get started with, I started by making sure that I could count down the maximum number of trump that could be available as the tricks progress. I only subtract cards played and don't deduct the cards in my hand until I play them. It's too easy to subtract them twice unless you get in to that habit right from the start.
Then I started keeping track of which trump are potentially available. I started by keeping track of the highest (boss) trump that could be out there. Even at the hardest, 7 out there and I have none (though I usually just ride these out and don't worry about counting) left-right-ace-king-queen-10-9 can rattle around in my head pretty easily and now I can forget about counting the 7 because I have the list that I can count if I need to.
I started expanding this once I got good at it, to the suits with aces outstanding. At worst, now I'm thinking suit-suit-suit-left-right-ace-king-queen-10-9.
Once I got really good at keeping this in my head, through practice, I noticed that I keep track of the suits in my head in the same order every time. Now, instead of tracking the suits with aces left, I switched to tracking the boss for each suit and all the trump ending up at boss-left-right-ace-king-queen-10-9-boss-boss (where the trump drifts around in position based on suit). It's easy to lose track of the off suit boss because they could be thrown out of order, off suit. (ie if the boss was King and the queen was thrown off suit, then the king was played in the next trick you could be waiting for the queen to come out ...)
10 cards out of the 24 available tracked helps a lot. I suppose I could shoot for more, but this is usually more than enough.
Depending on your memory, you could try tracking A-K-Q-J of each suit plus all the trump (watch where you count the bowers) ... this would give you 18 of the 24 cards, but I haven't gotten to this level myself.