Almost all of the rentals in Burlington have June to June leases but some apartments go on the market early spring and fall. This coincides with the UVM students leaving in late may and returning in August. Many students sublet over the summer or rent for two months while not living there. Competition for apartments near UVM and downtown is fierce. You will be competing with wealthy out-of-state UVM students. UVM is 65% out-of- state students that are paying one of the highest public university tuition cost's at 38K a year. Enrollment is also increasing every year with more and more students seeking off campus housing. The average 2 bedroom apartments near downtown/UVM are going for $1200 a month. One bedrooms/studios are going for $500-800 a month. Many of the apartments in these areas in my opinion are substandard for what you pay.
I would look into the South End (south of Maple st) or the Old North End (north of North st). With the South End being the better area. The Old North End is where most of the crime is located. These areas are much cheaper and are still within walking distance to downtown.
I am not sure what you are going to do for work or where you are moving from, as I have found Burlington to be very expensive in comparision to the wages being paid here and I am moving in less than a month back to CT.
I suggest you read this report from Burlington's Community and Economic Development Office. It is filled with tons of useful data on Burlington. I will add that unfortuntely most of it is negative.
Consolidated Plan
Good sites for apartments.
Burlington Apartments-Vermont apartment rental, relocation and real estate resource
vermont housing classifieds - craigslist