To expand a bit on what Steve says...
Yes, Audition is completely up to what you want to do and, because it's professional software, a ton more. The trouble is, this makes the learning curve much steeper than simple (but far more limiting) programmes. Just to give an example, on my own set up I can record 32 simultaneous sources (in multitrack) and set up multiple different feeds to monitor. With this versatility comes complexity though and I strongly suggest you spend a few hours with some of they tutorial videos available by searching.
To give you a quick overview of what you need to do to get started:
Go to File/New/Multitrack Session, set it up for the bit depth and sample rate you want, stereo or mono (likely stereo) and all that sort of thing, then give it a name.
Go to Edit/Preferences/Audio Hardware and tell Auditon to use your mic as the default input (you may need to go to your Windows Audio control and set it up there if you mic doesn't show up on the Audition list). Then set up the default output to be your computer speakers or headphone output.
On the multitrack screen, pick the track you want to record into (I'll assume you already have a music track that you've opened in a track) and make sure the input source is your mic--input is that box with the right-facing arrow in the area to the left of each track.
Click on the red "R" at the left end of the track to enable recording--just in the track(s) you want to record to, not the ones with existing material). If you want to hear your voice via the computer, click the small "I" next to the "R". This routes your input back to the output but, as Steve says, your computer might introduce latency.
At that point clicking on the red record button in the main transport controls should start you recording with playback going to your computer output.
Now, that's the basics but you have a ton of options (and a ton of shortcut keys if you choose to learn them). Why so complex for a single track? Simply, if you have the external gear to record multiple tracks at once then you need to be able to tell Audition which source to put in which track. Once you get going, you can create a "session template" if you want to use the same set up every time.
So...give that a try but I strongly suggest you watch some tutorials for more detail--and I'll second Steve's comment that, whatever DAW you eventually decide on, a priority purchase should be an external sound interface to bypass the one built into your computer.