Some places do not require any training at all. Just come in and learn the menu and be friendly to the customers.
Depending on what state and what company you work for, you may have to get certain certifications. Almost all servers have to get certified in basic food handling, which is normally provided by the job itself and costs nothing to you. The certification is normally good for at least a year and you can use it anywhere where it requires the certification, so it is not limited to just that particular job.
If your job serves alcohol and wine, then certain places may require you to be certified in the serving of alcohol/wine. Again this is normally provided to you by your job, they will let you know when you are hired.
These certifications are not hard and normally are common sense knowledge. Some jobs make it easy enough where you just go to a group meeting with all the other employees getting certified and they explain to you the procedures. Once the meeting is done, everyone will get their certificates without having to take any tests. Some areas/jobs may be a little harder, but it shouldn't be difficult none the less.