Alice Davies's reply:
Having a Fine Art degree is a great backbone to launch into any creative career.
A starting point for you could be to spend the day in book shops, interior shops, stationary shops and independent boutiques, and make a list of all the illustrators, graphic designers and design companies that you like the style of.
When you've got your list, contact each one and ask if they take on interns to see if you can get yourself some experience within this type of environment and grow your knowledge and experience from there. The most important thing above anything else I would say is that you are a nice, friendly person with a can-do attitude.
There will be others taking this approach too, so it's important that you get yourself noticed by having an interesting CV, perhaps done in the style of an infographic (see Pinterest for ideas). One person I know went to each company in turn and hand-delivered their CV and letter along with a box of doughnuts! Or better still, why don't you get your CV printed onto a box of doughnuts, which will surely get you noticed?!
Networking is also very important. Research your best options of being able to chat with influential designers, illustrators and small business owners. Find creative networking events, go to open studio events and look at what regular meet-ups these people might attend.
Get yourself a business card made up saying something like 'aspiring illustrator and graphic designer ready to learn and play!' and give it to everyone you meet to let them know you're looking to get into graphic design or illustration and that you're available for work experience.
Also you can't ignore Facebook. Put together a short post and message everyone you know. You never know who could be friends with a great graphic designer desperate for some admin help in exchange for showing you the ropes with the design skills you need.
I have confidence that it won't take long before you become part of a paid creative team. Go get 'um!
All the very best Rachel