Columbus apartment hunters are going to have the best luck in the months of May, June and July. If you wait any later than July to start your house hunt, you will likely start to see that inventories are shrinking and your options reducing. You may still be able to find a place, but will likely end up paying more or settling for a less than ideal apartment.
Finding a high quality apartment in Columbus isn't extremely difficult, but that doesn't mean you shouldn't pay attention to the details during the application process. Asking intelligent questions, such as how garbage pickup works and who is responsible for maintenance or yard work, will tell your landlords that you're going to be an excellent tenant.
Columbus is a fairly navigable city. If you're within the city center, Metra bus lines will get you within a few short blocks of anywhere you need to go. If you're traveling more frequently to the outskirts, you'll find that having a car is very worthwhile. Downtown is also totally navigable by foot.
Only one other museum in the State is larger than the Columbus Museum. Newcomers to town should really enjoy getting to know local history through the 14,000 items in the museum. Four parks within the city boundaries provide bike paths, public tennis courts, and even a fishing lake. There is also a paved trail that follows along the Chattahoochee, which is a wonderful way to enjoy a spring evening.