Cities or Countries nearby Bologna?

Casey, Would you fly to Phoenix if you wanted to explore Mexico or Canada?

I think what Casey meant was should he arrive in Bologna a week early by airplane and then travel by whatever means to other places. Or perhaps arrive in Europe a week early and explore other places before making his way to Bologna.

The country that is closest to Bologna is the Republic of San Marino. Switzerland and Austria are not terribly far. As far as places in Italy from Bologna you can easily reach Tuscany (37 min. to Florence). Other cities in Emilia (Modena, Reggio, Parma) are within reach, and so is Romagna (Ravenna, Forli).
The Marche region is close (Urbino, Ancona). The Dolomites are also not far. Pretty much anything in Italy north of Rome can be reached within 2-3 hours from Bologna.

Bologna airport has nonstop service to multiple other places in Europe; if you're willing to make a connection, you can go almost anywhere. You can easily see Emilia and Romagna cities as daytrips from Bologna, as Roberto said. I particulary recommend Ferrara, Ravenna, and Parma. For flight information, look at the Bologna Airport entry in Wikipedia to get an idea of cities it serves, and look at Skyscanner for schedules and prices. For train information, use Bahn or Trenitalia. Note that Bahn will not have prices for journeys outside of Germany, and Trenitalia will show faster trains in the future (I don't think all of June is loaded yet), but will only show regional trains for the next week. So, just pick a day within 7 days to see the schedule. I found that daytrips from Bologna often required leaving at 8:30 AM (earlier than I wanted) or 11 AM (too late as it wasted half the day), but had nothing in between.

Thank you all so much for the guidance! I will definitely check out those day trips.. and possibly Switzerland and Austria :) Thank you!!