North/North East towards Leh or Spiti Valley
This Road is only open 4 or 5 months a year between June and September approximately, depending on weather conditions. Estimate at least 2 days and 1 night to cover the scenic 500km between Manali and Leh. No permit is needed to transit this road (as of August 2019). Beware of altitude sickness, you'll be as high as 5300m of altitude. You can spend the night in Keylong, Darcha, Sarchu, Pang, Rumtse or probably other smaller camps along the way. Some of this are just seasonal camps, not towns. Don't expect hotel-like facilities after Keylong, but the bed spaces are cheap (around 3 USD a night) and offer blankets. Restaurants and tea stalls are more abundant.
If you are in Vashist just start hitchhiking (with your thumb up or waving) on the Leh-Manali Highway. If you are in Manali or Old Manali, cross the Beas River using the bridge in Manali and start hitchhiking right away. You can walk towards Vashist but it's not necessary to walk all the way out of town to get a ride. Bring warm clothes in case you end up in the back of a truck, because you'll be crossing 4 or 5 passes in total if going to Leh.
The first pass is Rotang-La, less than 2 hours after you exit Manali. Many Indians just go there for pictures and not further, but it's worth to take the ride and then hitchhike out of the pass.
If going to Spiti Valley, leave the asphalt road and take the dust road on the right at Granphoo (which is just a restaurant on the road, not a proper town). This is some kilometers before Keylong, right after you are done coming down from Rotang-La. The road to Spiti is rough and traffic is scarce, but it's doable. The first town is after the Kunzum-La pass, but you have 2 camps before that, where you can pay for a simple bedspace on the mattresses, rent a tent or pitch your own tent. Hitchhiking inside the Spiti valley is not hard but you have to be particularly patient, it's one of the most scarcely populated areas in the world. You only need a permit if planning to go beyond Tabo.
Go to the Riverside Bus Stand in the south end of Manali and start hitchhiking as soon as you can. If going to Kasol, leave the big road at Bhuntar (before entering the town) and go on the road along the Parvati River.
Other useful info
There are several ATMs, places to exchange money and reasonably good Internet in Manali, unlike many smaller towns in Himachal. Nothing like this exists after Keylong in the way to Leh.