Manchester Media Club is really great, and I'd recommend it to any students looking to go into the media. I've attended some great talks by them, and got really great work experience as a direct consequence of being a member.
If it's specific print journalism experience you're after, then the MEN is probably your first port of call. Don't forget the MEN subsidiaries too - you're more likely to get experience, and get to do actual work if you're interning on the South Manchester Reporter, or the Oldham Advertiser than the actual MEN. The Guardian also has a HQ in Manchester (it was originally the Manchester Guardian) - I don't have any experience approaching them, and I'm not sure how much content is produced in Manchester (I can't imagine there's much, if any) but it's always worth a try.
There's also a news associates HQ in Manchester, and if you can't get experience in the newsroom they offer open day taster sessions called 'Insight into Journalism'. They're trying to flog you a course they run (don't believe the spiel), but it's quite enjoyable and something to say you've done. Aaand, Ten Alps - Ten Alps own about a bazillion magazines, so even if you can't get work experience with them directly, I'm sure their website would be a useful one to peruse for Manc-based magazines.
And then you have the BBC, and Media CityUK - I'm not really interested in print, really, so I'm not sure what's coming to MediaCity in terms of print but I'm sure there would be companies there who could offer relevant experience.
And Student Direct, of course.