An AWS Welding Certification is usually sought after by welders with several years experience in welding. Though you can just register for the test and go take it at any time. The AWS Welding Certification must be re-certified every 6 months by filing the correct paperwork signed off by your employer proving that you are currently using the applicable welding standards.
You do not need any certifications to get an entry level weld job. Most employers issue you their own "weld certification" upon completion of their weld school. Those certifications are useless outside of the company that issues them.
A lot of the bigger manufacturing companies will hire you as a welder with your current credentials (high school welding class). They have their own weld school, usually a two week crash course. You will just need to be able to lay a half decent bead to pass their hiring/weld test which you will likely pass if you have any idea what it is like to lay a simple bead.
Many vocational schools, colleges and other adult education institutions offer welding. If they offer more than one class I would recommend going to the school and speaking to the instructor before enrolling. My local college has basic and advanced welding classes were the advanced class covers all the same material as the basic class and more, meaning that either class is suitable for an introduction to welding but you will get much more out of an advanced class.
Going the school route will let you learn about weld theory as well as hands on learning how to use Stick, MIG, TIG, Brazing, Plasma, Oxy/Ace etc. depending on the class. Again speak to the instructor before you enroll.
If you go to an employer weld school you will most likely only learn how to run beads with a MIG. Then if you get pretty good at it, have a little luck, and the employer has a need for someone with skills in other forms of welding like TIG, they may train you for that process as well.
Edit: Employers will pay for certifications and schools that they require you to have.