Lots of good advice already.... I'll add my 2c, some variations on themes already mentioned.
1. Start at Appalachia, take Fallsway to Sylvan Way to Howker Ridge Trail. This is an unrelentingly steep trail up through the woods, and then a lot of fun climbing up over the "Howks" to Madison. Then over to Adams, then down Airline, over the Knife Edge with great views into King Ravine, then right turn on Upper Bruin down to Valley way, but then quickly bear right an go along Snyder Brook on Brookside, and just keep to trails on either side of the brook all the way back to Appalachia.
2. Start at Appalachia, take Airline to Shortline to King Ravine Trail. Make sure to take the "Subway" portion of the trail, crawling under, around and through heaps of huge boulders on the floor of the bowl. Then up the headwall, very spectacular, but not at all scary. (I like the King Ravine Trail better than Great Gully Trail, the latter goes mainly through scrub, the KRT is more exposed with more scrambling). Then up Adams, over to Madison and down as in #1 or down Howker Ridge.
3. Start at Great Gulf Trail parking on Rte 16. Up Great Gulf Trail to Madison Gulf Trail (very steep, fun trail, not as tough as it reads in the book) to Adams then Madison (various choices around the summits, see map). Then down Osgood Trail (very nice ridge walk for quite a while) then back to Great Gulf Trail.
4. You can make a very big day by doing #3 but staying on the Great Gulf Trail to the Six Husbands Trail, crawling through crevices, over ladders, nice scrambling, to Jefferson, then over Adams and Madison and down as in #3. Do not try this in iffy weather as there's no good escape route back to your car on the long stretch from Jefferson to when the Osgood Trail finally drops off the ridge.
Have a great time. The Northern Presis can't be beat!