A set of steel wheels will pay for themselves within two tire swaps, i.e. by the time you put on winter tires this year and put the summers back on next spring, that'll cost as much as buying the extra wheels. Getting a cheap set of alloys takes twice as long to pay for themselves since they're usually about twice the price.
Even the cheapest winter tire will be far better than the summer tires on anything but dry pavement. Unfortunately you're not going to get very good performance on dry pavement, but that's the price you pay for skimping on tires.