You will get the best quality by downloading or capturing streams, but most probably you will need several programs for that like freeware StreamTransport, CooJah + Blader, Orbit Downloader / Grab Pro, shareware Replay Media Catcher, WM Recorder. Also note that on-demand streams can be downloaded much faster with stream downloading / capturing software.

Of course you can use screen video capture software (screen recorders) for recording streams, but you will have problems with your recordings if there will be any buffering when playing on-demand streams. As for live streams, in many cases it is more reliable to use screen recorders.

Since you plan to convert your videos, you will lose quality anyway, so it may not be that important to use stream downloading / capturing instead of screen recording.

Also take a look at bundles like Replay Capture Suite and WM Recorder Bonus Bundle. They include stream downloading / capturing software, a screen recorder, a converter,... It is much cheaper to buy a bundle than separate standalone products.

See also:
The best video stream recorder, stream downloader, stream ripper, stream capture soft
Stream downloaders, rippers vs. Screen capture software