Help! Need a photo touch up software!

Hello everyone this is my first post on the verge so please bare with me!

I am just getting into photography and I have realized how limited I am without any post modifications to photos. This has lead to many problems and the scraping of images that could of been salvageable otherwise! What I would like to know is what software is best for me to use. Please include the best free software ( if any worth mentioning) and paid for ( please don't brake the bank nothing over $100). My computer, I found came with Corel Paintshop Photo Pro 3x and I would also like to know if this is worth using. I have heard that it is very limited when it comes to RAW format and my Nex-5n preforms well in this format. Any advice would be helpful as I am very new to this and although I enjoy learning curve I rather not waste money or time on programs that are unnecessary! Thanks for your time!

-Alex