Hi Danielle. Painting tile is not a good idea. You might get it to stick for a while but it certainly no long term solution. I can understand that since you just bought a new house, you might have a lot of things that you want to change. So promise that you will not get mad when it starts to peel and that you will do it right as soon as possible and I'll tell you how.
Yes, wash it with Tri-Sodium Phosphate (TSP--available at any paint store or home center) and rinse. Sand it with a hard rubber or wood sanding block and 100 grit paper. You need to score the smooth, hard tile surface. There are some paints that are supposed to stick and you might check with a local paint store to see what they suggest and to have them add mildewcide to the paint. If this is a problem, I would go with an enamel.