Albany is a government town, and a college town. This means that the summer is definitely the time to be out looking for an apartment. June and July are probably the easiest months to find an apartment in this city. Try to avoid looking in August and October, as you will likely find supply much more limited at this time of year.
Many landlords in Albany are quite particular about who they rent to, so be sure to put your best foot forward. It is easy to overlook little details, like showing up on time and dressing neatly for your appointments. You don't need to show up in a suit and tie to view an apartment, but you shouldn't look like you just rolled out of bed either.
Albany is actually an excellent town to drive in, as parking is available and the traffic is light compared to larger cities. However, if you don't have a car you still have transportation options. The Capital District Transportation Authority offers service over a very large area, and should be able to get you close to any destination in the Albany region.
You can easily spend days looking at the architecture in Albany, as it contains many historic buildings. If you only see one building, make it the New York State Capitol. The New York State Museum is free, and has the largest collection of any state museum. Also free are the summer concerts held at Empire State Plaza. Lark Street is the most popular street for nightlife in the area.