Get the best grades possible in Growth and Development and your pediatric specialty courses. Demonstrate excellent time management, critical thinking and organizational skills. Those will make you immediately attractive.
By all means volunteer at the children's hospital; see if you can spend some time working with Child Life. That experience will help you learn how to interact with kids of all ages and developmental stages. Pay close attention to the teaching that Child Life provides for patients; your job will be easier if you can use similar strategies and reinforce what the child (and family) has already been told.
Develop a "mission statement" as to why you're interested in PICU/oncology but be really careful to avoid cliches and banalities. Draw on your personal experiences when you develop your statement; leave emotion out of it as much as possible though. Elaborate on how you manage stressful situations and have some examples you can use to provide detail. For example, when I started my nursing career 15 years ago, I could have written your post. My mission statement was that I wanted to work with seriously ill kids. My son had been cared for through several life-threatening events by some of the most skilled and compassionate nurses in the world. I wanted to give back by emulating those nurses. I felt I had something unique to offer and I played up my special qualifications to my benefit. My "inside" information has helped me through the years and made me a better nurse... and a better mom.
There is nothing wrong with speaking to the recruiter. That's their job! That individual will have good intel on what works and what doesn't for applicants. What do you have to lose? Nothing. Make a good impression and maybe you'll be remembered when your application lands on somebody's desk. Unless you REALLY thunder in your discussion, the worst that could happen is you won't be remembered, which isn't really a disadvantage.
There's probably no harm in chatting up your client either, if you go about it the right way. Start off with casual conversation. Say something like, "I think you told me once that you're a nurse at Children's. Am I remembering that right?" Ask if she likes her job. Ask her what she likes best and what's not so good. Tell her you're going to be starting nursing school and you're looking for your niche. It's not like you're going to be interrogating her, you can gather information over several visits. Keep it casual and light until you get a sense for how she's reacting. If it's positive, then you can get a little more detailed. Don't ask for her help though. If she offers it, take it, but let it come from her.
Best wishes for a successful entree into the wonderful world of peds nursing.