Day 1 – I agree with Rockoyster, going to Pilatus when you’re not getting to Luzern until 3 in the afternoon is unrealistic. If I read the schedule right, the last gondola on your arrival day departs on the ascent at 4 pm, which in theory you could catch once you get the bus to the proper stop and walk to the gondola base station. Worse, the last gondola for the descent leaves the top at 4:30 pm. (Starting the next day, those times increase by an additional hour.) See:
http://www.pilatus.ch/en/inform/timetable/
Since the two gondolas involved take about 35 minutes total to get to the top, you simply can’t do it. Even if you catch the 3:14 bus from in front of the train station, you still won’t get to the top until 4:19 pm, giving you enough time to turn around and go back down. So, you’d be better off exploring Luzern that afternoon & evening. Old Town is nice.
Day 2 – Certainly doable, Titlis would basically be a half day trip from Luzern, counting travel time, exploring time at the top, and maybe a short jaunt around Engelberg before heading back. I’ve not been to Bern but that also looks like a half day trip, or maybe a little more, based on travel times and time for exploring.
Day 3 – We went from Luzern to the Jungfrau area and back last October. Train from Luzern, through Brienz to Interlaken Ost, switched trains to Lauterbrunnen. Caught the bus to Stechelberg, gondolas up to Schilthorn and back down, bus back to Lauterbrunnen. Caught the train through Wengen to Kleine Schiedegg, stopped for lunch at the train station restaurant, train back to Lauterbrunnen, switched trains to Interlaken Ost, caught train back to Luzern going through Brienz again. Very scenic train, bus and gondola rides, not to mention the views from the Schilthorn and at Kleine Schiedegg. However, be prepared for a long day. Our day trip took 12 hours, leaving Luzern on the 8:06 am train and getting back at 7:55 pm. Fortunately, our hotel was close to the train station. I’m not sure you can fit much more into that day.