In Leh there are a number of shops that will rent motorbikes, mostly the Royal Enfield, still made in India today 350 and 500 cc model. Rents are fairly cheap, and if you are are used to old bikes and left hand side driving, it is certainly a great way to move around if short of time, and far cheaper than local taxis. Be sure to check your rented bike before you leave so that you don't end up getting stranded in the middle of nowhere. As always in India, drive carefully, as other drivers often lack caution.
Things to note
1. In most sections of the journey, the road are in a bad condition but in certain conditions the roads are literally non-existant. Bottom line is that BRO Border Roads Organisation has done a good job, with what ever little resources that are available, in making these difficult terrains accessible to vehicular traffic.
2. Though there are many mechanics in Leh who deal with many bikes, the availability of spares is limited. So before you leave please be sure to get your bike serviced also get all cables checked/ changed, set chain, get oils topped up, brakes inspected etc. and also carry all necessary spares cables, chain link, bulbs etc.
3. Make sure to carry the originals of all your bike's documents.
4. Glaciers tend to melt as the day progresses and flow at some places across roads. So be sure to plan to reach and cross these glacier melts commonly known as Nalas for example Pagal nala, Khooni nala, Whiskey nala, Brandy nala etc. during the earlier part of the day, when the flow is low and the depth depth of the water is still easily passable.
5. When you encounter a Military convoy, always pull over and let them pass. It might be a good idea to find out from the locals as to when the convoy goes uphill and downhill and try to time your trip accordingly.
For the traveler with a number of months it is possible to trek from one end of Ladakh to the other, or even from places in Himachal Pradesh . The large number of trails and the limited number of roads allows you to string together routes that have road access often enough to restock supplies, but almost entirely avoid walking on motor roads. See below in the Do section for more info.
If you plan to drive/ ride in to the Ladakh region in your own car/ bike,
1. Carry enough spares and all the required tools.
2. Try and learn basic vehicle maintainence before you start on the trip.
3. Carry spare fuel. There is a 380km strech on the Leh - Manali highway which has no petrol pumps.
4. You will need to get permits to visit certain places For example Khardung La
Ladakhi buses run from Leh to the surrounding villages. They are often overcrowded and generally disorganised and poorly run. Daily buses or mini buses run to Alchi, Basgo, Dha-Hanu, Likir, Nimmu, and Saspul; twice daily to Chemray, Hemis, Matho, Stok, and Tak Tok; hourly or more often to Choglamsar, Phyang, Shey, Spituk, Stakna, Thiksay.
The scenery would be magnificent at the pace of a bicycle, however one would need to be well prepared with full camping equipment. There is a bit less than 1000 km of paved roads in Ladakh. The Manali-Leh-Srinagar road makes up about half of that, the remainder being spurs off it. As such it's not possible to string together a loop, and the only route that would avoid backtracking would be to follow the Manali-Leh-Srinagar road. You would need to check the current situation and think carefully to decide if travling in Kashmir at bicycle pace is more of a risk than you want to take.
In addition to the paved roads there are some trekking routes that would be possible to ride a lightly loaded sturdy mountain bike on, perhaps hiring a horse and handler to take your baggage. Padam to Darcha, via Shingo La pass would be a good route for this, though you would still need to push your bike over the pass itself. Ask trekkers in Ladakh for more options.