Splinters are one of those common annoyances that we all have to deal with. If there is no complicating factors, simply removing the splinter with a pair of tweezers is entirely appropriate. You would only need to put a bandaid on it if there is bleeding
or oozing that comes after removing the splinter. The key with any injury to the skin is making sure to wash it carefully with warm water and soap, to make sure that the wound bed is as clean as possible.
Generally, that is adequate treatment. But it is always important - in this age of the many drug resistant bacteria that we so commonly hear about in the media - to keep an eye on the spot of the wound for a few days until it has healed fully. If you notice signs of increasing redness, warmth, pus or other discharge from the wound, then you might need to talk to a doctor
Additionally, if you notice that you have a fever
greater than 101.4F and have any of those signs, or red streaks up your arm, you would need to talk to a doctor. Generally, tweezers and careful watching are the perfect answer, but speak to a doctor as needed.