If your phone is locked to Vodafone (i.e. it came with a Vodafone SIM pre-installed) it won't work on any other network unless you pay to get it unlocked. (Almost every market seems to have a stall with a guy who can do the job for you at a reasonable price).
If you purchased your phone 'SIM free', and then put a Vodafone SIM into it, it will already be unlocked and you can insert any other provider's SIM into it straight away.
Once you're in possession of an unlocked phone, simply call in at any large branch of Asda and speak to the guy on the sales desk where they sell phones, CDs, DVDs, memory cards,etc. He will know all about switching your phone across to Asda's own network (which is probably the cheapest PAYG service and uses EE's network). However he won't be as arrogant as the staff in mobile phone shops nearly always seem to be. (Before I get accused of sexism, by the way, I'll point out that I'm using the male gender simply because it always seems to be a man behind the technology counter in every branch of Asda. Similarly mobile phone shops seem to be staffed almost exclusively by men. I've no idea why that is!)
He'll be able to explain about the savings you can make by purchasing 30-day bundles and things like how to get your Vodafone number transferred across to Asda's service or, of course, you can find all of the information here:
https://mobile.asda.com/