This part of the Himalayas is a bit restricted in places as it's a huge border area. Not that you should be worried as generally, it won't affect you at all, in fact, you're going to be welcomed with open-arms by perplexed locals pretty much everywhere you go.
Some places in the vicinity do require extra permits and permissions sorting before you're allowed to go there though. Along the main route you'll be fine with just the Inner Line Permit for the Spiti Valley, but if you plan to go off-piste it would be a good idea to do a little research on your route before you leave.
Have a read of this and it might help. It also might not.
Below are any places we know of that require extra permits.
- Karzok Lake permit is very easy to get in Leh. If you're coming from Shimla then the permit for this may be available from Keylong (TBC).