In the popup window, you can name your project and choose a destination. There are optional templates you can use (like for podcasting, film sequence, etc). I would stick with none for now. For the bottom three, the default settings are fine, but just in case; set the Sample Rate to 44.1 (44100), Bit Depth to 32, and Mix to Stereo.
At this point, it depends on your project. You can start recording your own audio or import existing audio and edit that.
To import audio, you can double click in the files panel, or go to File/Import
To record audio, make sure your microphone is plugged into the computer or a soundboard. To double check your microphone, click on Adobe Audition/Preferences/Audio Hardware. Your microphone/soundboard should be listed as the Default Input (or listed in the dropdown menu for input). The Default Output can be Built-in Output (your computer microphone).
On the Track Controls, click on the R on the track you wish to record on (R on Track 1).
To start recording, click on the Red Dot in the Transport Panel. You will start seeing a waveform building in the Timeline. Click on the Stop Button in the Transport Panel to stop recording.
Right below the Files Panel, there will be a panel for the Media Browser and Effects Panel. As you apply effects, pay attention if you are applying an effect to the entire track or just a clip.
Selecting a clip will open the Essential Sound Panel on the right. Here, you can tag clips as Dialogue, SFX, or Music. You can then apply specific effects to that tag. (If you are missing any panels, you can open them under Window at the top).
Dialogue: To even out speakers’ volume, select the clip, add the Dialogue Tag in Essential Sound. Under Loudness, click Auto-Match. Another way is the select a clip, right click and select Match Clip Loudness.
Dialogue: Under Repair, click on Reduce Noise. This applies Denoise to your clip.
Dialogue: You can apply Dynamics and EQ under Clarity in Essential Sound or apply them under the Effects Panel to clean up your audio. To apply an effect in the Effects Panel, click on the arrow at the end of the row, select a category, then effect.
The other options under Essential Sound mostly auto-match loudness for music and sound effects.
The tool bar at the top has a Razor Tool to cut a single clip or everything under that clip. You can then use the Selection Tool to remove the unwanted clips.
Like Adobe Premiere Pro, there is a difference between saving a project and exporting a project. Saving the project saves your project files, not your final file. Exporting mixes down your timeline and renders it out as a MP3 file.
To save your project, click on File/Save As. This will save your project as a .sesx file (Adobe Audition project file). In Multitrack, this will show up as a folder wherever you set your project up.
To export your project, click on File/Export/MultiTrack Mixdown/Entire Session. In the popup window, you can name your file and choose a location. For Format, I would use MP3, but you could choose other options. The other settings are blocked out, but they match your project settings. When you are ready, click Ok.