Hi Courtney,
Thanks so much for visiting the Disney Parks Moms Panel with your question.
I suggest that you consider the weather and the crowds when making this decision. In the first few months of the year, it will likely be cool in Florida, with temperatures averaging highs in the 70's Fahrenheit. These temperatures tend to rise starting January 1st on through spring. The best way to avoid crowds is to avoid times when children will be out of school - so avoid New Year's Day, Martin Luther King, Jr. Day, and Presidents' Day. Toward the end of March crowds might begin to pick up as Spring Break season begins. Given all of this, I would suggest a vacation toward the end of February or beginning of March. My first trip to Walt Disney World with my daughter when she was an infant was in February. The crowds were light, and we wore long pants and light jackets most days.
As to a resort for your vacation,
Walt Disney World offers many resort hotel options at a variety of prices. The one that you chose will depend not only on price, but on which theme appeals the most to your family. If you are looking for the most inexpensive options, you can filter the options to show only Value Resort Hotels. Depending on your budget, you might also include Moderate Resort Hotels. For a family traveling with an adult or two and a 1-year-old, either of these options will work. If your decide on a Value Resort Hotel, I really like Disney's Art of Animation Resort, which immerses guests in the animation process. For a Moderate Resort Hotel, both of Disney's Port Orleans Resorts - both Riverside and French Quarter - are lovely.
I hope that these thoughts are helpful for you, Courtney. Have a great time planning your vacation, and please do not hesitate to ask if you have any further questions!