i'm brushing my teeth in a second. if this swollen gum crapola goes away i will brush my teeth more often.
Gums don't bleed when they are healthy. It is important to keep them claen, just as it is your teeth. The real trick though is cleaning below the line where your teeth are covered by your gums. You can actually put a small pick deep into the area between your gums and teeth if they are healthy and they will not bleed.
Now the horror ...
FLOSSING!
I never did this at all, until I was in my forties and I did not really know how to do it properly. I have a nasty story to tell, after you get all healed up, about what happens if you do not clean below the gumline on a semi-regular basis. I haven't lost any front teeth, but I have lost two molars from improper, incomplete care.
Most likely, your gums are swollen because of a bacterial infection below the gumline and it might be OK, but you would be smart to pay particular attention to that area with a good anti bacterial mouthwash, over the next few days. Some flossing to get under the gums can also help. Yeah, they are sore and it will hurt, but no amount of gum pain is worse than losing a tooth. After you get some anti-bacterial in there you can often make them feel a little better by using a solution of warm saltwater, holding on the area. This will help a small bit, but it will often open up the capillaries and allow some additional, healing bloodflow to teh area, also.
Are your teeth loose in the affected area? If so, you could have a serious periodontal problem.