Replace the stock speakers, definitely, I am about to do that myself. But first, start out by replacing the HU (headunit) which it sounds like you already want to do. A lot of people here will recommend Alpine units, which are well-built and usually match the interior lighting colors. Other brands I personally like are Blaupunkt (see my sig), which isn't often mentioned in car audio conversations, Eclipse, and Pioneer (they also often match the interior lighting if you look). Apart from that, look on sites like crutchfield.com and etronics.com to get started, and certainly go to a local audio store to try the decks out yourself. In terms of purchasing, I highly recommend buying online, especially through ebay, where you can get some awesome deals.
For installation, you really can do it yourself, with the radio removal tools and a factory wiring plug adapter you can pick up at your local Pep Boys, Napa, etc. After that, its pretty much just connecting wires and sliding your new deck in.
The speakers should be next to replace, and you can search on this forum for recommendations. I don't have my own yet, so I wouldn't call myself much of an expert, but there are lots of posts about them. The same would go with amp and sub choices.
Good luck with all your audio upgrades.
