Don't restart! Don't change the difficulty to peaceful! (Though, of course, you can.) Part of the fun is getting obliterated on your first few nights :D
Here are some strategies:
- Fight! The skeleton will circle you, so circle it so that you 'spin around each other' as you get closer to it. This is better than running straight at it, though you can do that too. Once you land your first punch, which will knock the skeleton back, keep moving towards the skeleton and keep punching! You only need to hit it 10 times for the kill.
- Flight! Strafe (by pressing A or D) as you run. Get to higher ground, and avoid other monsters. Dig a hole straight down into the dirt and cower inside of it.
- Freeze! Find creative ways to let the skeleton kill you until the morning, at which point you'll be more or less safe.
- As a last resort, change the difficulty to easy (by pushing escape) until you get the hang of things.
After you survive, you should build yourself a shelter, and find a way to get light. There are a number of tutorials for 'surviving your first night' (you can look this up on youtube), but they'll just spoil your fun. All you really need to know is: a log becomes a plank, a plank above another plank becomes sticks, and coal (which you find and mine) above a stick is a torch.
If you have some blocks with you, a good strategy is to put a 3-block-high wall (or pillar) between you and the skeleton. If it's two blocks high, the skeleton can't jump over it, and if it's 3 blocks high, the skeleton can't jump-and-shoot over the wall. Keep this wall between you and the skeleton. Once the skeleton is right on the opposite side of the wall, slowly edge to the side, so that you're peeking past the corner. The skeleton will slide sideways to peek at you (and try to shoot you, but you should be safe). Once you see the skeleton, hit it. It'll bounce backwards. Repeat until it's dead. This works much better in caves, where the skeleton can't circle.