Just came back from a wonderful vacation here, where the entire purpose was to hike in these mountains. Here are some observations that I think will be helpful as more people come to this region who are not necessarily experienced hikers (like me).
Physical fitness - you need to be physically fit to enjoy the hikes here. I am middle-aged, not overweight, and would say I am slightly above average in this category. Fitness is especially important if like me, you live in a low-elevation environment and cannot easily practice in similar alpine environments.
Mental fitness - these hikes will also test your mental health. There will be times you will be challenged, one minute you are doing fine and the next you are facing exhaustion or a scary situation. You need to be able to talk yourself out of the situation and evaluate what needs to be done.
Difficulty - my opinion is that there is no such thing as an easy hike here for the average tourist.
Danger - you should have the mentality that you are going to a dangerous environment. Tourists and experienced hikers die here regularly (over 80 dead in just over a 30 day period this summer). If you are not prepared, it could be you. You need proper hiking garments, shoes, layers, plenty of water, and food. You need to be prepared for any kind of weather. At one point I was doing great on a morning hike until a had a very steep climb that coincided with the hot sun coming out. I felt pretty exhausted with all that coming together. Luckily I had what I needed on hand to get me to the next hut where I could regroup and cool down.
Itinerary - I see a lot of people on this forum talking about coming here for 2 days on their whirlwind European or Italian vacation, right between their stays in Venice and Rome, or Florence, or wherever. If that is you, you probably need to think about all the above points more than others. The mountains (and Europe in general) are not a playground.
When to visit - for the hiking season you should really be coming here between Jun and Sept.
August - People said not to come during August, but I found it to be just fine. I was hitting the trail around 6 or 7 in the morning and encountered few people at those times. Get up early and August is not a problem.
Ortisei - people act like this is the only place to stay. There are better places to stay, like almost any other town in my opinion. I stayed in a nice residential area in Brixen and it was perfect. No other American tourists.
Aple di Suisi - was nice but my least favorite of all the places I visited. Not a must-see at all.
Lago di Braies - not an overrated destination, but you have to get here very early or late.