Answer:
It really depends on what kind of a spider has bitten you. If you can capture the spider that bit you, or kill the spider without damaging its body, your doctor can check what kind it is.
In the United States, only the Black Widow spiders can kill human beings, and then usually only children or infirm adults. So if you brought in the body of one of these spiders, your doctor would know exactly what to do. There is even an antivenom for these bites that can be used if the bite is bad enough. If you know you have been bitten by a Black Widow, it would be best to see a doctor fairly promptly. The younger the patient, the more important it would be to get to the doctor promptly. If you are in Australia you probably know what the Sydney Funnel Web spider looks like, and if you are in S. America you probably know what Phoneutria (armed) spiders look like. They are both hard to miss.
The only other U.S. spider that is a major problem is the Brown Recluse. These spiders are sort of "anonymous-looking," i.e., they are brown, medium sided, etc., like many harmless spiders. So it would really help to have the culprit in hand if that is what bit you. Their bites cause local tissue death, an effect that is manifested relatively slowly, so knowing that you were bitten by this particular kind of spider could be very useful in guiding treatment before any severe symptoms developed. (Spiders in the same genus occur in S. America and can be more dangerous.)
Spiders can best be caught by putting a drinking glass down over them and then sliding a 3 x 5 card under the edge of the glass, turning it glass right side up, and taping some kind of cover on the glass.
The bites of other spider may be painful (about like a bee sting), but the pain will go away. The only big thing to worry about would be the same kind of shock that some people get from an allergic reaction to bee stings. If somebody is going into shock it is clear that they need to be seen immediately.
Some bites are suspected of introducing germs into the wound. Spider fangs are sharper and thinner than hypodermic needles, so it may be useless to apply antiseptics since the wounds will seal as soon as the fangs are withdrawn, but it would not hurt anything to dab on some Isodine (buffered iodine) or other good antiseptic.
An ice cube wrapped in a wash cloth and held on the wound might make you feel better, and it also might let the body deal with the venom bit by bit instead of all at once. I think that treatment is useful for bee stings. It has been a long time since I have been bitten by a spider (my fault) even though I study them and "play with them" frequently, but I think that is what I did the last time I pinched a spider and got a defensive bite.
If you are only noticing local pain, then probably all you need to do is to clean the wound, treat the symptoms, etc. If your whole body is reacting somehow (e.g., cramps), then it would be safest to get medical help. If the local wound seems to be getting worse instead of getting better, it probably would be a good idea to be seen by a doctor too.