Hey Matt, My name is Mike and I play/captain the team at University of Illinois-Chicago. It's good seeing another Chicago team pop up! I'm in a similar situation to you as this is one of the first semesters that we as a team are trying to attend tournaments as a full team (re: not Rom playing solo). My first advice would be to try to talk with student organizations and see if they can sponsor quizbowl. We got kind of lucky and UIC Quizbowl is sponsored by student organizations so they pay for most of our tournaments. Also, I would try to set up practices in the first couple of weeks in January by advertising around campus. I'm not sure if you guys have a buzzer set to practice on, but if you don't you could always swing by UIC for a practice.
Tournament wise: There are a few good tournaments coming up in February that are "Regular Difficulty" that may be good for a starting team, though some may be fairly difficult. Those tournaments are NAQT SCT http://naqt.com/sct/hosts.jsp
(which is going to be hosted by UChicago), Penn Bowl at Northwestern http://hsquizbowl.org/forums/viewtopic.php?f=8&t=12331&p=231152&hilit=penn+bowl#p228916
(Though the field is full and a waitlist has started so It may be difficult to get in) and ACF Regionals at UIUC http://hsquizbowl.org/forums/viewtopic.php?f=8&t=12511&view=unread#unread
. There is also going to be MUT (Minnesota Underground Tournament) http://hsquizbowl.org/forums/viewtopic.php?f=8&t=12309&start=0&hilit=MUT
that will be a more appropriate difficulty for newer teams that will be hosted sometime in March.
Anyways, If you have any questions shoot me an email. I hope to see you guys at future tournaments!