Not all ASDAs do on-site processing, but it can be great, they will do a couple of films while you do your shopping, for £3 including scan onto CD. They do tend to over-sharpen their scans, which are quite low res at 1200 dpi, enough for 4*6" prints, but it's a service that's hard to beat. Max Spielman and Snappy Snaps on some High Streets are a bit pricier.
BTW all of the above are only for C41 (common colour negative) film, not slide/transparency or black and white film.
My favourite choice for quality 35mm C41 processing is Photo Express in Hull, £5 per film (less 50p per film if you quote your TP username) plus £1 for return postage for process and a nice 2000 dpi scan, much better than ASDA. They are currently averaging 2.9 days for me from posting to dropping on my door mat. Postage out costs an annoying £3.20, but you can send quite a few films for that, if you're careful with packing!
Look in the Film and Conventional sub-forum on here for more hints on processing black and white or slide film. However it can get quite a bit more expensive, specially scanning.