My first "job" was more of a small gig as I just searched all around the internet for anyone publishing comics and needed help with layout/typesetting. Eventually, I was able to find a very small publisher who was looking for help with their book layout. I was paid $100 for it. Wasn't much but it helped.
Then I worked for the Digital Manga Guild as a typesetter which, as many probably know, was pretty terrible. It was a lot of work for almost no pay (I worked on 4 books I think and was paid just over $100), but again, it helped a bit and was another thing to put on my resume.
The place I found to work part time was as a salesperson at an art "gallery". Well, we liked to call it a gallery, but it was mostly just selling random, cool art stuff. It wasn't a bad job and I ended up working there for 5 years.
I kept searching around for other jobs and eventually did find a paid "internship" at a place that made class rings and they were basically looking for people to do some grunt work of redrawing things in Illustrator. It also helped a lot and allowed me to work around my current schedule so I had two part time jobs for a while.
That lasted about 7 or 8 months and then the internship ended. Even though they were hiring and did say we would have a chance to be hired, they didn't hire anyone from the internship... they ended up hiring a completely new person... with the same amount of experience and a really annoying voice so I don't know what was up with that.
No biggie since right after that I was finally able to get my first full time graphic design job! The application process was shady though since they did a "test", but it ended up being a legit job just at a startup with no idea of what they were doing. I was still able to work there about 3 or 4 years and survived several rounds of layoffs until my time finally came.
Finding a job after that, now that I had experience, was a nightmare. I sent out SOOO many resumes and received ZERO back. I just don't even know how that was possible. When I was job searching while working, I actually had a phone interview and the person told me I was one of two or three people even qualified for the job I applied for, but now I got nothing. I think it's because designers are a dime a dozen, employers think anyone who has touched Photoshop can do professional design work, and everybody already "knows" somebody or knows someone who knows someone and just hires them through word of mouth.
Which is how I got my current job... My old boss was laid off before I was and was now working at a new, more established company, and was able to get me in.
So, yeah... mostly it is knowing someone and getting lucky. Don't expect simply sending out resumes to get anything back, but your resume should still look PERFECT.