Gimp is quite good and it has a host of tools. There are also plug-ins like UFRaw but you will not need that with Jpeg. The other free software is
Picturenaut which is not going to help you with what you have now but is very good at doing HDR.
On the commercial Photoshop CS5 is really good but on expensive side but Adobe also does Elements which is a cut down version of Photoshop. There is also Paint shop pro similar to Photoshop but a lot cheaper.
Schools use both Paint shop pro and Photoshop so there is a lot of guidance on both programs and most Pros will use Photoshop and Photomatrix the latter is more for HDR.
I think Gimp is great however the guidance in comparative to Photoshop is some what lacking and it takes some time to work out how to do things.
However although there is a tool to remove noise to be honest it can only do so much and once lost very hard to get it back. Over exposure is also a problem once lost it's gone. Under exposure not so bad although it tends to get grainy you can do something with it.
I would start on a copy of course with Colours/Levels and make the image span to whole useable spectrum after that try Filters/Enhance/Un-Sharp mask try about Radius 1 and Amount 1.30 this is for gimp with Photoshop it will be completely different.
If you do use Photoshop I only have CS4 then I would try loading as a RAW file. Yes I know it is a Jpeg but the RAW part of Photoshop is amazing and using the correction brush to lift and drop the exposure levels of set areas is really good. You can do same in main program but not as easy to use.
My version of Paint shop pro is rather old version 7 and I am sure now much improved. Good cameras let you save images in RAW and the better software will allow you to work in RAW if you ever get a good camera you may want this feature.
In general its not what the software can do but how easy it can do it. OK there are some amazing features like the predictive clone stamp of CS5 but how much you would use it.
However I have take pictures in the Pub with my phone and combined them with Photoshop Panorama to very good effect.
You may also try using black and white one can bring out some bits with filters in black and white you would not believe.
The same applies to HDR you don't need a D-SLR to take HDR you can still get same results with a compact and where you have control of the focus you can even use the stack to get depth of field I did it with my microscope.
Have fun and down-load Gimp it is a really powerful tool.