Q: How does one go about writing for children’s television?
A: As a grizzled veteran of the showbiz wars, I am sorry to report there are as many ways to get into animation writing (accidentally or on purpose) as there are ways to get into a San Pedro bar fight. There are many paths and a lot of them are overgrown with weeds and hard to find.
Unlike in the real world where you go to school, get a degree in something, and then have a reasonable chance of getting a job in that something, careers in animation (and entertainment in general) seemed to be fueled by an often imbalanced combination of sheer luck, perseverance, and talent. Oh, and knowing someone that already has a job in your targeted field helps too.
A few tips to point you in the right direction:
1) Personal contacts. If your area college offers classes in entertainment related writing, this would be a good place to start. Classmates with similar interests and goals often end up in a position to help each other. Also, network with any friend or acquaintance who works in the industry. Even if they're not working in your specific area of interest, they might know someone who is and can introduce you or pass on a script. I know it all sounds kind of mercenary, but it's a hard truth. I'm not saying to "use" your contacts (this will quickly gain you a reputation as an annoying leech), just let it be known that you're looking for animation writing gigs and would appreciate a "heads up."
2) Be ready to back up your networking by having a writing sample (or two or three) ready. These serve as your currency in entertainment writing. When you do end up making contact with someone who counts, they'll ask, "What have you got to read?" Sitcoms are the common choice in this area (yes even in animation). I think producers like them because they aren't as long and involved as screenplays and are a quick gauge of whether you can handle character and dialogue (my expertise is in the "funny" shows, I don't know what kind of scripts they read for the action cartoons). I recommend having several samples as a particular producer might have a particular dislike for a particular sitcom. Also try not to pick one of the "big" sitcoms. Even though "Friends" was a very popular show, producers were sick of reading samples for it because everybody and their uncle had written one. A popular up and coming show that looks like it has legs (will last a while so you don't have to write another sample right away) would be a better choice. SIDE NOTE: Get a production copy of a script for the show you are writing and make sure your script is formatted to look EXACTLY like theirs. Deviation from the accepted format brands you instantly as an amateur and greatly increases the chances your script will be tossed. There are script sources on-line and at your local bookstore.
My experiences as a writer of "Histeria!" illustrate some of the above. I was an actor at the Groundlings Theater in Los Angeles. Just to keep busy, I started writing sitcom specs. It turns out that the producer of Histeria (Kids WB) was a Groundlings fan and put out word through the grapevine that he was looking for writers. I followed up on the lead, sent over my specs, responded positively when they ask me to write a couple of trial historical sketches (for which I was paid), and got the job. It turned into a two and a half year gig. Could I have planned it? No, but I was ready for the opportunity when it came.
I hope this helps you or at least gives some food for thought. When you do end up being a showbiz big shot, don't forget to throw your old pal Roger a bone (networking!). Good luck!