It depends on the particular illness. With some diseases you can spread (be contagious) before symptoms appear, others you don't become contagious until after the symptoms appear.
Viruses and bacteria have evolved to reproduce and migrate in different ways. Some spread by contact and can survive in open air for a long time--these could possibly live on external skin or be on kitchen counters, desks, pens, pencils, etc. and be transported to a new host when the new host comes in contact with the infected area. These are probably also most likely to spread before a host shows symptoms.
Other types require a moist environment to survive and cannot exist in dry, open air. These would need to be spread via bodily fluids (saliva, sperm, blood, etc.) and are often spread through coughing or sneezing--hence the symptoms are there in order allow the organism to spread to a new host.
Edit (per Matthew Read's comment, here are some links):
The Wikipedia entry for Influenza describes the method of transmission for that disease. It confirms that a person with influenza is contagious before they are symptomatic.
The Wikipedia entry for Transmission (medicine) lists multiple routes of transmission for a variety of maladies, from HIV to athlete's foot.
Edit II (per Borror0's comment):
The US Center for Disease Control (CDC) offers a number of notices and other articles aimed at the general public and health care professionals. In an article on preventing the spread of the flu virus in child care settings, a brief explanation is given as to when an infected person can be considered contagious:
People with influenza can potentially infect others beginning 1 day before symptoms develop and up to 5 days after becoming sick.
Other articles provide information on the transmission route for a variety of viruses. For instance, an article about the Norovirus says:
Noroviruses are transmitted primarily through the fecal-oral route, either by consumption of fecally contaminated food or water or by direct person-to-person spread.
Another article explains the transmission of the rabies virus:
The most common mode of rabies virus transmission is through the bite and virus-containing saliva of an infected host. Though transmission has been rarely documented via other routes such as contamination of mucous membranes (i.e., eyes, nose, mouth), aerosol transmission, and corneal and organ transplantations.
Meningitis can be caused by a bacterial infection or a viral infection. According to the CDC, viral meningitis is contagious only after symptoms appear and most stop being contagious when the symptoms go away. The same article also offers the following information about bacterial meningitis:
[S]ome forms of bacterial meningitis are contagious. The bacteria can mainly be spread from person to person through the exchange of respiratory and throat secretions. This can occur through coughing, kissing, and sneezing.