Audacity is the standard for anyone working on sound using Linux or windows based computers. There are other applications out there for specialist use and a small number of video applications have good sound editing. Start with Audacity.
Audacity is a 5.3 MegaByte download and uses 16 MB on your disk. A few minutes of high quality sound files will use more space. Plan to use a GigaByte per hour of high fidelity sound. You will need several GB spare when editing so you can store multiple copies of files until you are completely satisfied with the sound. Save your files with different names at every major step in the editing process. You then get the option to backtrack and try several variations of an edit. As an example of disk space usage, I recorded an interview that filled one GB. After several experimental edits and some variations on some experiments, I could not backup the project to a DVD because 4.7 GB was too small. I backed up the project to a single layer Blu-ray disk, 25 GB, and almost ran out of space.
Download Audacity from audacity.sourceforge.net.
A large screen helps you edit audio because you can see the controls easily across a wide selection of time. You want an accurate mouse or large graphics pad for quick selection of time points on the display. Large audio files work best with lots of memory and fast disks. Most modern processors are fast enough to edit audio.
You need a fast processor for video editing. Given the low cost of good quality digital cameras with external microphone connections, you could make it a habit to record all audio using video cameras so you can use the visual cues from the video.
You also need a fast processor if you move up from two tracks to 5.1 or 7.1 sounds. You need a fast processor for some special effects. You will need a very fast processor if you decide Audacity is not flexible enough and you step up to Ardour to apply multiple special effects at the same time.
Good headphones help editing because you can listen to the sound without distractions from the environment.
Ardour is a step up from Audacity in both complexity and capability. Learn the basics of audio editing using Audacity then try Ardour if you need more. Ardour is easily installed with a GTK based user interface by the Ubuntu Software Centre. Ardour accepts LADSPA and LV2 plugins from various sources, giving you the choice of many effects. Effects can be chained together to create whatever you want. Edits can be undone, giving you the freedom to try lots of changes without damaging your audio file. Ardour would be the next application I would learn if I was editing audio for a few hours each day instead of a few hours each year.
If you spend a lot of time on audio and video and want to use only Linux, consider Ubuntu Studio. Ubuntu Studio is Ubuntu preloaded with Ardour version 2, Gimp, Blender, and several latency reducing modifications for Linux.
LMMS is an alternative to Ardour. LMMS does not have the unlimited undo feature of Ardour.
Transverso DAW is another alternative to Ardour and does have an undo feature. I like the look of Transverso DAW but would probably choose to use Ubuntu Studio if I had to professionally edit lots of audio tracks. Transverso DAW talks about automatically applying effects to all the tracks in an album, something you might do if mass producing pop or techno or easy listening albums.
Jokosher looks simpler than Audacity and has an add-on module facility that might be able to bring in the the extra features you need. There is an instrument oriented menu instead of a special effects menu. You could build musical tracks in Jokosher then add special effects in Audacity.
Ubuntu Linux installs PiTiVi for video editing. If most of your output is video, you use only one or two sound tracks, and only a few effects, PiTiVi could work. When you start overlaying multiple music tracks and special effects to merge into a video, you could run the audio through Audacity to use the extra special effects.
Blender is a popular animation composition and rendering tool with some audio processing. You could perform simple audio work in the Blender video editing section. You would probably build audio special effects in Audacity before adding them to the video in Blender.
A good book helps you get started by explaining things in a logical sequence. Some parts of Audacity look obvious while others are a mystery until you experiment extensively. The first two books in the following list are the best choices.