My first research experiences were more exploratory than anything else. I worked in a plant ecology lab, essentially watering and weighing the samples. And then I worked in a plant cell wall lab, where I ran PCR and gels. Those were very low-level positions and were for me to discover what research I was interested in pursuing later at a higher level.
After that, beginning of my 2nd year, I started my first 'real' research experience. I started working on a publishable project in that lab toward the end of my 2nd year, and I'll finish the manuscript for publication at the end of my 3rd year/graduation. As for the writing on this project, I have a document with paragraphs of ideas, but my PI will integrate those as she sees fit and do the majority of the writing/formatting.
My actual undergraduate thesis is not publishable, though it is higher level field work. The research is simply not novel enough to be worth publishing.
I don't think anyone expects you to be great at writing papers, when you're just leaving undergrad. And I agree with TakeruK that you should focus on the research, because it will help drive writing the paper. To prepare for grad school, my biggest piece of advice would be to read as many papers as possible, in different major journals in your field. Get an idea of how things are worded and formatted, and try to implement that in your own writing when you get to that point in your research.