Empty the entire thing, including pulling up the cruddy rug. Rent a power washer and give it a good cleaning. Seal and stain the concrete floor. Home Depot, etc. has kits for this. Get some pretty plants in terra cotta containers, tables from yard sales, etc. to display them on, hanging flowers from hooks, nice patio set, etc.

Removing the carpet will make it so much easier to keep clean. Remove all things that don't make you happy or need storage elsewhere. Paint in a cheerful color on the walls and furniture, more potted plants, and some pretty wall decorations will go far to spruce up your patio.

The old carpet may have spilled food left in it which attracts bugs. It won't cost any money to remove it, so go ahead and do that. The flooring underneath could be painted and sealed, if you like. So could the woodwork of the porch. Check out yard sales, Craigslist or Ebay for some outdoor furniture. Some plants around the outside of the porch could be added for color and texture and to soften the hard lines of the porch. These are all things that could be done as a family project and perhaps even be fun.