Doing the math, it was ~1900m (6200ft) vertical in ~14km (9mi). We did it in less than 4 hours but we were really moving.
At the top, you'll be able to see the Sphinx, the cross, or join the other expansive network of trail systems.
Dress accordingly, one of the times I did this hike, all was good, except for about 1hr... We were caught by a nasty rain, hail and thunderstorm. We were on the high plateau, with no coverage and were nearly washed away by the flash floods.
Mountain Bikes are available for rent from Sinaia, so go ahead and rent one for the day from Montania if you like. Montania also gives the option to hire experience guides that can accompany small groups of people on the mountain bike trip, so feel free to ask them about this.
You can go on bike to Cota 1400 with no problems at all. There are several other trails that you can take. They are very well marked, so don't be afraid that you will get lost. Just follow the signs and you will be fine.
I also heard that Montania now organizes mountain bike trips to Omu Peak, the highest peak in Bucegi. I did not have a chance to ask them about it since this is a new thing that added after my last visit to Sinaia, but feel free to do so if you are there.
Sinaia's ski slopes are situated at 2000m altitude. They offer you the best high altitude ski in Romania. The ski season starts in December and lasts sometimes until May. If you do not have your own ski equipment, you can rent it from a lot of places in town. Daily and weekly rentals are available for tourists.
The slopes can be accessed from the city center by taking the cable car to Cota 1400 and then a second cable car to Cota 2000. You can also go to Cota 2000 using the 4-seats express chairlift.
You can also go to Cota 1400 by car, but taking the cable car is way more fun and much faster.
The ski slopes are divided on the two sides of the mountain: to the east side, towards the city, you get the most spectacular slopes, the black ones. The slopes follow the cable car line and there is a great variety of off-piste possibilities. All the slopes end in the same point, which is the cable car station at Cota 1400. The slopes are not for the beginners, though!
On the west side of the mountain there is the great Sun Valley. The Sun Valley has a milder aspect and here you can enjoy the beginners and intermediate skier trails.
Dorului Hut is a good starting point for cross country and ski touring. The experienced skiers can explore the Dorului Peak and its surroundings. Touring routes can be made towards Vanturis area (in the south), Bolboci area (in the west) and Blana-Laptici areas (in the north).
There are lots of places for rock climbers enthusiasts. I admit, I am not one of them, but I have seen several people doing this while I was in Sinaia, Busteni and Predeal.
From what I heard, the walls in Sinaia are the easiest ones to climb. The "real thing" however, can be found in the Costila Region, only 30 minutes hiking from Busteni. Busteni is easily reached by car or by train from Sinaia. It takes about 10-15 minutes to get there.
Stop by the tourist office once you reach Sinaia or Busteni and ask for a map with all the available trails for rock climbing. They will gladly provide you with one!
Sinaia is often referred to as a "ski town", although from experience, the locals tend to ski Predeal or Poiana Brasov, but opt to stay in Sinaia, which I believe is better known for its 'apres ski', with many bars and pubs.