We suggest you begin your career by working as a web development contractor.
Web Development contractors are hired on temporary contracts (usually 3-9 months long) by companies who need temporary help on a project or who want to try out developers before committing to hire them.
It is usually easier to be hired as a contractor. While it is a temporary position without the benefits of full-time employment, contract positions make up for that by paying you more money than their full-time employees.
By looking for contract jobs first you are more likely to find work, and you are able to get a variety of experience. If you spend your first year on two different 6 months contracts you will have built up some great experience and added two positions to your resume.
Do not worry about having a lot of short term employment on your resume. There is no stigma in web development towards short-term employment. It is understood that many people work on contracts.
Many companies use contract positions to test out developers for full-time positions. Sometimes companies will hire 5 contractors with the intention of keeping 3 for full-time employment at the end of the contracts.
These are called contract-to-perm positions, and they give you the length of the contract to prove you are worthy for the benefits of full-time employment (if you want it).
Check for contract Web Development positions in your area.