Well, I'm in the scorpion cave-thing. It says I have 10mm rounds x2, but after killing maybe 3-4 scorpions it says that I'm out of ammo. Any solution?
Also, the doc in the Sandtown told me to wait till the morning before I talk to him, how do I rest/sleep?

Pressing z will let you select the options to wait/rest.
For ammo, go back to vault 13 and talk to the overseer on the third floor. Convince him that you need supplies and then talk to the guard (Not the water guard) by the supply closet he is guarding. There is alot of ammo in there and also a shotgun.

There should also be some ammo in the caves. I think it was in one of the branches towards the bottom of the screen. But if you ran out before you got that far, it might not help you.
Although you might be able to run past the Scorpions in combat mode and pick it up, then load your gun, then start shooting at the Scorpions. That would be "Rad" ;-)
Kind of like that scene in Untouchables where the baby carriage is going down the stairs in the train station and they toss the loaded gun and shoot the bad guys and save the baby.
To pull this off, you need to press "A" on your keyboard to enter combat mode before the scorpions see you and go aggro. Then you can use all your action points to run past them. You might need to run near them and then lure them away from the ammo, then run past them and pick it up.
Or you could just recruit Ian in Shady Sands and bring him along. Let the scorpions sting him, NPCs can't be poisoned. Once you pay him to join you, you're allowed to use the steal skill on him with automatic success to manage his inventory. So you can take your money back from him and grab some ammo too.

The guns don't reload automatically.
Right-click on the gun to cycle through the modes... one of them will be a "reload" action.
Also, the scorpions are best beaten with targeted single shots. Burst mode wastes way too much ammo.
If I were you I'd make sure to get a hunting rifle before going to the scorpion nest - makes it far easier. Heard there was one hidden somewhere in a bathroom in Vault 15...

I always get Ian before I go, mostly obtaining the necessary caps by stealing Shady Shands blind. Anyway, even without him, make sure you use the targeted shots with your pistol (right click on the weapon at the bottom middle of the screen until you get a targeting icon, then click the target). If you run out of bullets, go with the spear, someone in shady sands mentions that the scorps don't deal well with light, so if you light up one of your flares the scorcs damage goes waaaaay down.

.,,,

Depending on your skills and stats you can concinve Ian to join for a piece of the action instead of paying him

Knife is also effective against scorpions even if you have low skill. As long as your agility is high you can take a swing and then use all your action points to walk away.

It can take time, but that way you can clean the whole cave without using a single bullet.