Starting at Rifugio Aurenzo follow the path to Cappella degli Alpini, before arriving at Rifugio Lavaredo (which can be closed depending on the season, so check beforehand). Exactly to your left, you'll see the three peaks rising up above the landscape, while on your right, the other jagged Dolomiti mountains.
From Rifugio Lavaredo, turn left and follow the path upwards towards Forcella Lavaredo, a nice view point on Tre Cime di Lavaredo. From here, you can take breathtaking pictures over the valley and towards Rifugio Locatelli.
Your next destination is Rifugio Locatelli. From Forcella Lavaredo, you can hike up two trails, one of which is steep and the other -- easier and well-trodden, both of them are leading to the same destination; however, if you are confident in your hiking stamina, I recommend taking the least trodden path, as it has a few spots with epic views. Nevertheless, since the trail is quite steep, it might be dangerous, so if you're not confident in your hiking skills, better take the easiest trail.
Once you've arrived at Rifugio Locatelli, take a break on the terrace, have a drink or some food or both, and enjoy the stunning scenery of Tre Cime. If you have time, walk around Rifugio Locatelli and explore the area. There are a few bunkers and tunnels left from WWI, as well as other interesting photography spots to get a good perspective of three peaks. There's a nice Laghi dei Piani below the Rifugio that you can also take a glimpse at or shoot from above.
Hike up to the nearest peak, which will take about half an hour -- along the road there is a beautiful view of Refugio Locateli and three peaks. Otherwise -- you can go along the left path and climb 200-300 meters to a little moutainscape tunnel from which you can see the three peaks.