i was in the same position as you this past may, graduated and didnt have much to my name.
graduated may. studied for dat on my own with textbooks, review books, practice tests, and software. took DAT July 28th. I lined up and did all of my "100 hours" of shadowing over the next two weeks. First week of august completed my application including shadowing projections - which i did actually meet 🙂.
While studying you will be bored enough to fill out the actual application, get your transcripts sent (as early as possible after june 1st) and have your LORs in, and choose your schools!!! review the school requirements as some take anatomy. make sure you read the forums to see which DAT material is the best.
speaking of LORs. some schools require one from a dentist. throughout my shadowing, i found one really cool dentist and had him write me one which then was submitted a bit later to the 2 or 3 schools who required it.
check out topscore and crackdatpat right now also. i think they are good for 6 months, and if so, i would recommend purchasing it during this semester and take a few practice tests (of which you can forget, and retake during the summer). crackdatpat will allow you to familiarize yourself with some weird logic type games. then when the summer comes, youll know what to expect and you need to do.
register for the DAT ~2 months in advance.
are you familiar with how the application cylce works?