Book a four day minimum trip. Two days for Katahdin plus drive out and back. Go to BSP website and book sites ASAP. Anytime after July 31st through mid September is prime. Weekends are real popular and the park will be full. There is a parking pass system for day hikers who want to climb Katahdin, basically ithe three trailheads are Roaring Brook, Katahdin Stream and Abol, once you pick a campground decide what trails you take up to the summit. There is a way around this issue if you are camping in the park but it entails getting up around 4 AM.
When you get there pick the best forecast and do Katadin that day. Plenty of other hikes for the other day, just dont wear yourself out on a side hike as Katahdin is a long day. Chimney Pond is great but it requires a backpack in, there is cut off time to head up there so you may need to get there early otherwise backpacking up to Chimney and then hiking to up Katahdin is long day. If you are first timer I would suggest the KSC or Abol for three 3 nights. The park is highly regulated, follow the rules and have a great time, dont understand or follow the rules and you will not have a good time as you wont be in the park very long. Abol and KSC are more family oriented campground while Roaring Brook is somewhat more quiet. CHimney Pond is the ultimate spot in the park but most folks line up in January to get a spot, otherweis its the lukc of the lottery and there are tricks that those familiar with the reservation system know that makes it weighed against you.
The trails are long and rugged, but ten year olds hike them. Its tough to recomend the best hike as many on VFTT are "connoiseurs" and all have their favorites. Hit a good day and all the summit trails are great
The AT (Hunt) trail is great but it requires some bouldering (pulling oneself up over boulders), Abol is one long steep slide. If you hit good weather you will be blown away with the views and terrain, quite unlike the whites. The Chimney Pond trail from Roaring Brook to Chimney Pond and then up to the summit via Cathedral and then down via the Knifes Edge and then Helon Taylor back to Roaring Brook is the one of the two most spectacular day hikes, its a long day, if you dont mind extra hour or two subsitute in Hamlin Ridge Trail in place of Cathedral. Realistically for a first timer, its real hard to pick a bad trail (although the Saddle is best used for a bad weather escape. THe knifes edge is not for someone with a fear of heights and there is at least one borderline technical spot. In other places you have to hug a vertical wall with a several hundred foot drop, plenty of footholds and handhold but its unnerving to some. Dont even consider it on less than a perfect day.
The brothers or doubletop are good picks for the other day.
One hint, plan to hike early preferably at or around sunrise, the summits can cloud in in the afternoon and you do not want to be up on the plateau in a thunderstorm.