All businesses in the City of Los Angeles must obtain a business license. You will probably need to register your new business with the Franchise Tax Board and the Internal Revenue Service. You will need employer identification documents if your business will employ others. I have included a link below to a government web page regarding starting a business in Los Angeles.
I hate to be the bearer of bad new, but you may not be able to conduct business from your apartment. Standard residential leases preclude using the apartment for business. Therefore, selling clothes out of your apartment is likely not an option. Food service is regulated by the City, County and State. Before you engage in a catering business check the site below and do further research on the legal requirements.
The other response provided excellent advice. All Los Angeles businesses require a license from the City, and payment of City taxes. There are also income taxes due to the state of California and the Federal government, and property taxes due to the County of Los Angeles. You will also probably need a re-seller's license from the Board of Equalization.
And as Mr. Gentry noted, your landlord and apartment rules may have restrictions on the types of activities you can conduct out of your apartment.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.