For editing your podcasts (or any audio clips) a very good, and recommended tool is the free, Open Source, and highly-featured Audacity, which you can download from the Audacity website. Audacity is available for PC, Mac and Linux.
NB: Audacity, for patent reasons, can't directly export MP3 files - you have to download a separate MP3 encoder file known as LAME and place that on your machine - see the Audacity FAQ for details and links to the LAME downloads for Windows and Mac.
Other editors you could use are:
As an hour-long lecture could result in a MP3 file of some 40Mb, your students will thank you for editing it into smaller chunks, if you can. You should also add at least minimal metadata (see below) when prompted.
There are some tips and tricks below which you might find useful. There's also links to 'screencast' tutorials on Audacity Audacity on the el@n (e-learning @ Nottingham) site.
To create a label to mark a position in a sound file, click in the waveform where you want the label to appear then choose Project | Add Label at Selection, or press Ctrl-B. A red flag should appear in a 'Label Track' below the waveform, so then type a meaningful but short name for the marker, eg 'part1'. If you're playing back you can also pause the playback and choose Project | Add Label at Playback Position , or press Ctrl-M. The labels have no effect on the audio clip - they are purely visual aids to editing.
Whilst you're working on the audio file, it's quicker to save to Audacity's own file format via File | Save Project or Save Project As..., then when you've finished editing and have the file as you want it you can export it to MP3 with File | Export as MP3.... Saving in the Audacity format is helpful as it preserves non-audio data such as labels. Exporting to MP3 is relatively time-consuming, particularly for large files.
By default, the recording kit will produce a stereo file, which when opened in Audacity will appear as two tracks, one above the other.
Changing the sample rate: by default, your recordings will be 'sampled' at 44kHz, which is CD quality sound. You can reduce file sizes by 're-sampling' the file to a lower rate, 22kHz being satisfactory for speech. To re-sample, click on Project Rate in the bottom left of the Audacity window and choose the desired rate from the popup. Any rate below 16kHz will sound poor, with 8kHz being 'telephone quality'.
Changing the sample format: the default "sample format" is "32-bit float", which means that 4 bytes will be devoted to each sample. You can change this to "16-bit" without any loss of voice quality, which would reduce the file size as only 2 bytes will be used per sample. To change this, click the down arrow on the top left of the track display by the track title (the Track Pop-Down Menu), and choose Set Sample Format.
Converting stereo to mono: for podcast purposes, mono is fine for recordings. To convert a stereo clip to mono, you'll need to split the stereo track - click the track pop-down menu, and choose Split Stereo Track. This should give you two tracks with their own title menus, one of which you can remove by clicking the X in the top left of the track window. You can then set the other track to mono via the the pop-down menu, in which Mono will now be enabled.
It's important, for yours and your student's sakes, that you add at least minimal metadata - artist and title - to your podcast tracks, and ideally other metadata elements - eg genre ("speech"), album title. If you don't add any metadata, then when the tracks are downloaded to a media player such as an iPod, they'll appear on the player's display as "Unknown title" by "Unknown artist", which is pretty unhelpful for the listener, particularly if you've produced multiple podcast episodes.
In Audacity, when you export a track as and MP3 file, you're prompted to add metadata. This is optional and many skip it, not realising the usefulness of metadata or even what it is. You can retrospectively add metadata in Audacity in Project | Edit ID3 Tags. The metadata is embedded in the MP3 file.
A good way of viewing the metadata (?) you've added, and seeing how it'll look in a media player, is to download the Apple iTunes software then import (File | Add file to library) your track into it. As a comparison, try importing a music track from a CD or MP3 and seeing what metadata that comes with.
You can edit song metadata in iTunes - right-click on a track, choose Get Info then click the Info tab. However, this'll only edit the metadata in the imported copy, not the original. A better tool to use is the curiously-named but excellent and free foobar2000, which allows you to set metadata for multiple tracks (amongst many other useful things it can do). Use File | Add Files (or Add Folder) and select the tracks the metadata of which you want to edit, select one or more tracks in the foobar window, then right-click and choose Properties.