Is it possible...
 

To convert VHS footage into video files (avi or whatever)? If so, what programme/cables/equipment do I need?

Yeah, you'll need a video input card with the right inputs fro your VHS player and then some software.

Probably best googling it as I've never done this.

You will need a video capture card.

I just got one with dual digital tv tuners for $60 - cheap as chips.

They come bundled with the software for capturing from just about any source

http://www.newtech.com.au/img/p/2866-487-large.jpg


Or you can get a usb card, for around $45 that comes with the software and cables all ready to go - for you this is probably the best option

http://www.techbuy.com.au/images/large/P134662.jpg

I have done this via a VHS/DVD player combo that allowed you to record the VHS right onto a DVD. Downunder's method is probably cheaper unless you can find a used machine on eBay or something.

Well I have one of those space saving PCs, the draw back of which is that you can't do much in the way of upgrading/adding graphics cards etc.

Is that USB option just as simple as plugging into a port or is there a card involved?

The VHS/DVD combo option is an alternative but probably a pricey one.

What I'm looking to do is put all of old wrestling VHS's onto a portable hard drive.

The USB is just plug into a spare USB port and off you go.

Sounds good.

Now lets say I'm looking for one of those on a UK website, what would I search for?

USB Capture should bring something up.