Although stings and bites from bugs are in general just bothersome, with no lasting ill effects, there are times that they can cause more serious issues.  If there is an allergic reaction or the bite results in an infection, this can be serious and even cause death. 
There are symptoms that let you know when there is a problem like an allergic reaction or an infection and the attention of a health care professional is necessary.  If your child has previous problems from insects that sting and bite, it is of course imperative to let the child’s care giver be aware of possible danger.

In the case of a bee or a wasp that stings, the area should be washed carefully with warm water and soap.  This should be done about three times a day until healing is complete.  The bee stings and the stinger has a venom sac attached so removing the stinger as fast as possible helps to prevent infection.  The wasp does not leave a stinger in the skin so they will sting more than once if they can.  A cloth used as an ice pack and applied to the sting area for several minutes will help, along with ibuprofen or acetaminophen for the pain.  An oral antihistamine will help with itching also.  Before giving any medication to your child be sure you have consulted the physician.  Calamine lotion or corticosteroid cream on the area of the sting can also help. 
If the sting happens anywhere on or near the mouth medical attention should be sought immediately to prevent any swelling that could close off the airways.  Also seek immediate help if there is a rash or swelling and if the pain continues for an excess of three days because this would mean an infection has set in.

An allergic reaction can be life endangering to if there is a tightness in the chest or throat, wheezing or breathing difficulties, fainting or dizziness, vomiting or nausea or the tongue, face or lips swell, seek help immediately.
 
Bites that are from spiders are handled pretty much the same way.  Use warm water and soap to wash the area three times daily.  Use the cold compresses and give ibuprofen or acetaminophen for pain.  An antibiotic ointment will help to fight against any possible infection and if it is on a child, make sure their hands are washed often.  If you believe the spider was a poisonous one like a black widow, watch for any signs of allergic reactions or infections and seek medical attention immediately.  In fact, for a child even if you are not sure it was a poisonous spider, seek medical attention right away.
In the case of a tick bite, it will be necessary to take tweezers and pull the head of the tick out of the skin.  Save the tick by putting it in a jar of alcohol to kill it and take it to the physician when you go for treatment, especially if there is a child involved.