How can I get new clients for my graphic design firm?

Talk to your school. You did go to school for graphic design now, didn't you?

At least someone taught him how to ask this and similar questions on dozens of websites. Since months.

Advertise in your local alternative newspaper and any trade magazines.

Also look into any independent comic conventions in your area. Take your work there, rent a table, and hope some of your work catches the eye of someone whose looking to hire out a graphic designer.

Hello Friends,

Marketing and Advertising.
You need to send out brochures and self-promotional packets to advertising agencies and any other businesses that could use your work. You need to place ads in magazines that are aimed at the markets that can use your work. You get business cards made with a good example of the type of designs you do, and give those out to everyone you meet.
1) Find out how much other graphic firms charge and keep your prices a bit lesser than theirs, but make sure it's still profitable.
2) Organise an official store opening event - invite people you know, business neighbours, even throw in an advertisment in local newspapers/websites and open it to the public announcing that your offering 'special discount prices' to the first 100 customers or something similar.
3) Being a graphic design firm, you'll need to get in touch with some printing business (I suggest you keep a few instead of one). They usually get plenty customers/walk-ins who might be interested in getting more than prints made. Ask if you can have flyers put up at their place or at least ask them to recommend you to potential clients.
4) There's also a good market in online orders - many people want logos, websites, etc. created and would pay good money if they feel you're work is worth it (create some good examples to showcase). Some would even give you bigger orders and are willing to pay postal charges.

Thanks And Regards.
Agili Ron