Any time you see "cartridge" or "sealed" to describe a bottom bracket that means it's not serviceable - you throw it away and buy a new one. Or in some (expensive) cases, replace individual bearing races. The wikipedia Bicycle Tools page has more information. I'm using amazon below to give you an idea of prices.
You will need the appropriate tool to remove the cranks, and another to unscrew the cups that hold the BB in place. Both are readily available but cost about the same as getting new BB fitted. If you want to accumulate tools you could use this opportunity to buy them but it's something that many people decide to get done in a bike shop because it's rare and much easier with a proper work stand.
The problems with a worn BB will be obvious without dismantling anything. If you pull a sealed BB apart it's unlikely you'll be able to reassemble it. The key is how much slop there is in it. If there's too much, it needs replacing. Too much is a judgement call.
The measurement that matters is the spindle length. This is easy to measure once you have it out of the bike but hard to do before then because the cranks cover the bit you're measuring. If you're lucky it will be written somewhere that you can see without pulling it out, but usually the label is somewhere that is inside the BB shell when it's on the bike. Note that they come in 2.5mm increments, so you only need to measure to +/- 1mm.
The easy way is to take it to a bike shop. It's not instant even with the right tools, but you need to apply considerable force to get the cranks and BB off the bike, making it an ugly operation at home. It can be done, but it's not fun. They will pop the old BB out, read or measure the spindle length then try to upsell you from the bottom of the range Shimano BB to however much you're willing to pay. Realistically it's something you'll probably not detect, but the UN54 ones are IMO a good balance between really cheap and just good enough to justify the extra cost (they cost about $US20). You can spend $US200+ on a Phil Wood if you like (they are much shinier).
Also, most bikes come with British BB's, and yours is one of them. Other options are French, Italian and ... read more here (that describes the thread and diameter of the cups).
Edit: one final note. Bottom brackets commonly use both left hand and right hand threads. Again, pedalling the tool forward tightens the part, pedalling backward loosens it. At least with British and Italian BBs, with French they're both right hand threads. French BB's exist to teach us why left hand threads are used on BBs that work.