Why don't you look at the web site that has more details? The MOFA site is mentioned often here.
VISA
More details than you can shake a stick at.
Get a certain work visa, and you can only work within its limitations. Humanities Specialist/International Relations has a wide berth, perhaps the widest.
Want to switch to an engineering visa? You'll first have to be qualified. Again, read the regs on the MOFA site.
Bear in mind, too, that some of the regs are mere guidelines, not hard and fast rules, but that's a bureaucracy for you. I'll be happy to try answering any specific questions if you have a scenario in mind, but please give some background info on yourself.
Also, my last questions are; if you have a 'working visa', can you work part time in shops etc?
Depends on the work visa. The HS/IR type will probably allow that easily enough. This scenario sounds a little odd, though. Care to explain?
And also, do you have to be sponsered for a working visa (I ask this because some websites say you need sponser, but I can't find it on MOFA.)
The answer is yes. It may be implied on the MOFA site, more than directly stated, but one trip to immigration will clue you in. Or look at this site and read what each application form for a work visa says (about what employer must be listed). It's the first item.
http://www.moj.go.jp/ONLINE/IMMIGRATION/16-1-1.pdf