Saint Paul has rental suite availability that is on par with the rest of the country. However, the city has fought urban sprawl for many years now, and to find an apartment in the location you're after will take longer than it might in other similarly sized cities. Start looking in April or May to position yourself for success.
When you're applying for an apartment in Saint Paul, try to use as many local connections as you can in your application. In other words, if you are enrolled at a local University, employed by a local company, or have rental references from other Saint Paul and area landlords, it will be very helpful. If you're coming into town for the first time, you'll need to make sure your application is that much better.
There are a couple of public transit options in Saint Paul. There is a local bus service that runs throughout the city and also a light rail line that connects Saint Paul to its sister city of Minneapolis. Both sides of the Twin-Cities are very drivable, but you're going to want to be equipped with good winter tires for the latter half of the year.
For some local flavor to your shopping, take a stroll down Grand Avenue. Saint Paul has a great food and drink scene, and there are a few neighborhoods such as Grand or Snelling Street throughout town that have numerous options within a short walk of each other. The Turf Club hosts a lot of local bands and is popular with the younger crowds.