USB mic

You'll experience a HUGE improvement by going with a USB mic. The Blue Yeti mic is highly recommended, is at the low end of your budget, will record in stereo, and includes a mic stand.

Another option may be to use something like the Zoom H2n, so you have a portable recorder and it can be used as an USB mic too - more flexibility.
Also using a standalone recorder without plugging it into the laptop may be better anyway in some situations, e.g if the fans/ventilation of the laptop make too much noise.
The Zoom has switchable X/Y, M/S stereo and surround mics - I haven't used it a lot, but it seems pretty good for its price.

I opt for the ZOOM. You can use it as USB audio interface (microfone) and as stand alone device for recording as well. I use a 6N with good results.

Wow, the Zoom looks fantastic!

Zooms are great. I have the basic Zoom H1 but probably the H2n or H4n would be better.
Besides the on-board mics and USB connectivity you can also use Zooms as a line-in for recording from another source (mic with pre-amp, a sound board, etc.)
Very versatile and good build quality.

V

Thanks for the responses. As follow-up: What I found on the Web agrees with the comments above - Blue Yeti and Zoom H2n seem to be great value for money for voice (think podcast) as well as music recording.

In the end, my geeky side won out however, and I went for the Audio Technica AT2020 microphone (an entry level studio mic) and the Focusrite Scarlett Solo USB interface. In addition to the mic input, the Focusrite interface also has an input for Line-in (for keyboards etc) or instruments (guitar, bass, etc).

I’ve had loads of fun playing around with this for a few weeks now. The sound quality is great. (I never knew the clock in our living room was so loud!) But, to be fair, I can’t make any meaningful comparisons to other setups.