I have a bedroom wall outlet plug that is currently switched on/off by an existing wall swtich. I want to set up that switch to turn a new ceiling light on/off and leave the wall plug controlled also by the wall switch. Is this possible and safe to have the existing wall switch control both the outlet plug as well as a new ceiling light? If so, what type of re-wiring do I need to complete? Thanks in advance.

You can run a light right off the socket. If you do this, the light and the wall cannot use more amperage than the switch is recommended for. Most are 15 amps or so. A hundred watt bulb uses less than 1 amp (.833).

Install ceiling box at the desired location. Run a 2-wire cable from the outlet to the ceiling box.
At the outlet, connect the white of your cable to the existing white wires. To do this, you may have to connect all of the white wires with a wire nut and add a pig tail. Use the pigtail to reconnect to the outlet. If both halves of the outlet are controlled by the switch, connect the black wire of your cable to the black wire going to the outlet with a wire nut and add a pig tail. Use the pig tail to reconnect to the outlet. If only one half of the outlet is controlled by the switch, connect your black wire to the wire connected to the switched half with a wire nut, add a pigtail. Use the pig tail to reconnect to the switched half of the outlet.