Applying to a state college with a bad transcript?

Right now I am coming to the end of my career at a community college (next semester should be my last), and I plan on applying to a few small state colleges around my area.

When I first started out in college I messed up hard. I failed several classes, dropped my gpa all the way down to a 1.12, and even got kicked out of my program for two semesters. Since my struggles I have worked hard to get my grades up and get back on track with my education. I have now retaken all my failed classes, and I have brought my GPA up from a 1.12 to a 2.47. I plan on retaking even more classes this fall to raise my GPA even higher. Will the state schools that I try applying to hold my previous struggles against me? Will some schools not accept me because of what happened? I hope the schools that I apply to would see that I improved dramatically and actually note that as being good. I am just nervous that some schools will hold my bad GPA's and failed classes against me, and reject my application because of it.