Screen repair patches from local hardware stores are fine for a quick fix but they look horrible. Like everything else in life, patches have a time and place but these are not a long-term fix. Like a band-aid they stop the small cuts but eventually will need a permanent replacement.
When it comes to screens there are many options available to choose from. Starting with the standard fiberglass mesh, pet mesh, pool and patio mesh and Super Screen. Many options for your budget.