I wouldn't do the Armstrong/Gothics thing. Not that it's not nice, but since you are (a self admitted) list person, and it is a far trip to get there, you will probably want to maximize your peaks while minimizing your trips (classic max-min problem)
You may decide to camp at JBL, and get the wolfjaws, armstrong, gothics one day, and the rest of the range the next day. You're too new to the adirondacks to know for sure if you may want this. So save that one for another day.
I'd suggest trips that can't easily be combined with something else.. Like Allen. Well, maybe not for your first
I wouldn't do any of Marcy, Phelps, Tabletop, Skylight now, either. Again, once you become more familiar with the area, you'll combine them the way you want, according to mood/abilities/seriousness of finishing a list, etc. I wouldn't do a trailless peak for your first time here, althought I'd still like you to join me (and my axe) at 4:00 AM for Street & Nye tomorrow.
So, what does that leave you? Colden, Giant/Rocky, Big Slide, Cascade/Porter, maybe Whiteface/Esther, and maybe Algonquin, Wright, Iroquois.
Whiteface is NY's version of Washington... Road to the summit. I wouldn't suggest that now. Wright through Iroquois would be a long day. If the weather is great, and you're all feeling good go for it.
Cascade/Porter are the opposite. If feeling tired, they're an easy climb with so-so views. You may want to consider them as a morning climb on a day you're heading home.
Giant/Rocky are nice. A less distance, lot of climbing.
Colden is a longer walk, but less climbing. Interesting rocky-double summit. Some fun rocks (kids like fun rocks.. at least mine did). Returning via Avalanche Pass gives more fun rocks.
Big Slide via the brothers provides lots of views as you climb the three brothers. You'll have increasingly nice views of the Great Range (Gothics et al) as you get closer to Big Slide. The view of Big Slide from the third brother is impressive. I remember my kids reaction at the time: "We're going THERE??? Wow!" But my guys were younger than yours.
I trust you have a guide/map. Any questions, email me, or find me around the Loj parking at 4:00AM tomorrow.