We used to keep the vaccines (in their boxes of course) in drawer organizer bins in the fridge/freezer, with the name of the vaccine written on the end part of the bin so we knew what was where. You might be able to keep public/private vaxes in separate sides of the fridge to keep them separate too.
One other thing you might do, just to make your life easier, is know the CDC's pink book forward and back. The office likely has a copy, but if not, the health department surely does and might let you make photocopies for your personal use. If all else fails, you can find it used on amazon.com (at least you can right now).
There is a bunch of good information about vaccine spacing, what vaccines to give when, and contraindications for vaccines, as well as accelerated schedules for those kids that need to be caught up fast for whatever reason. Knowing those things might help you anticipate, 'ok, this kid is coming for a 6-month WCC, he'll probably be getting XYZ vaxes' and you can plan your time accordingly.
That book was my savior when I worked at a community health center. I got index cards, read every chapter, and wrote the highlights on the cards about each vaccine.
You'll also get faster with time....but the previous posters are also right. You shouldn't be rushed. It's worth doing right. Hope this helps.